Fifteen-year-old Gauff - a breakout star in 2019 - is in the same quarter of the draw as 23-time grand slam champion Williams, meaning the pair could meet in the last eight.
Gauff has never faced Serena but stunned her sister Venus Williams at Wimbledon last year.
The teenager took a step towards a first clash with the 38-year-old as she saw off Viktoria Kuzmova 6-3 6-1 in the first round on Monday.
"I felt confident, I feel like I'm moving well," Gauff said, with Laura Siegemund or Alison Van Uytvanck up next. "I had a good off-season, and I think that's showing in my matches."
Williams was not in singles action for the first time this year, however, as opponent Svetlana Kuznetsova withdrew, meaning the American must instead play Camila Giorgi on Tuesday.
The WTA great was still able to thrill the Auckland crowd, as she joined the soon-to-retire Caroline Wozniacki in the doubles, beating Nao Hibino and Makoto Ninomiya 6-2 6-4.
"I had a blast," Williams said. "It's so good to play with her - it's like a dream come true."
Wozniacki, who is calling time on her career at the Australian Open, added: "It's amazing to be able to share the court with Serena."
Elsewhere in the singles, Caroline Garcia recovered from a set down to beat Taylor Townsend, while Eugenie Bouchard, Daria Kasatkina, Tamara Zidansek, Jil Teichmann and Jessica Pegula all also advanced.
There were upsets at the Brisbane International, meanwhile, as Danielle Collins crushed Elina Svitolina 6-1 6-1.
Home hopeful Sam Stosur got the better of fellow former major winner Angelique Kerber, recovering in a tough first set to claim consecutive tie-breaks.
"This is the way I want to be all the time on court," Stosur said. "I put more pressure on myself than anyone could ever do."
Ajla Tomljanovic came through three sets, as did Barbora Strycova against Johanna Konta, with Sofia Kenin and Alison Riske also winning.
Aryna Sabalenka cruised through in Shenzhen, while Garbine Muguruza was made to work for victory against Wang Xinyu.
Wang Qiang and Elena Rybakina ensured all four seeds in action on Monday made it into the second round.
Nadal will return after close to a year on the sidelines at the Brisbane International, which runs from December 31 to January 7, as he targets a bid for a 23rd grand slam title at Melbourne Park later in January.
The 37-year-old has not competed since this year's edition of the Australian Open, having suffered a hip injury during a surprise second-round defeat to Mackenzie McDonald.
Nadal attempted to recover from his injury before the start of the French Open in May, only to be forced to withdraw from that tournament and undergo season-ending surgery.
The 'King of Clay' has said 2024 will likely be his final year on the tour, and while Rune accepts it will be difficult for Nadal to recapture his best form, the Dane is excited to see how he fares.
"It's great for tennis that Rafa is coming back," Rune said. "It brings even more fans to the tournaments and more excitement for the sport. I think this is good.
"We're going to have Novak [Djokovic], Rafa, [Carlos] Alcaraz, [Jannik] Sinner and more guys for the Australian Open. That's going to be super fun and it's exciting to see how it's going to go.
"You see some videos where he's practicing hard, but of course it's always difficult to come back and play after so long. But I'm excited to see it."
Jack Draper was the last player to lose to Nadal, going down in four sets in the first round of the 2023 Australian Open, and he is keen for a chance to avenge that defeat next year.
"I mean, I think it's amazing for tennis. Obviously, he's put in a lot of hard work to be back on tour, I'm seeing lots of videos of him training very intensely," Draper said.
"It's just good for the spectators, the players, that someone of his calibre – one of the greatest of all time – is back playing.
"I hope I get a chance to play him again because I think I was one of the last people to play him. It'll be amazing to have him back on the tour."
World number 32 Alexander Bublik echoed those sentiments, adding he was excited to see who will have the honour of being Nadal's final opponent if – as expected – he retires next year.
"Of course, it's very exciting that Rafa is coming back," Bublik said. "He's a legend of our game.
"But I just discussed it with Holger… I'm more excited to think who is going to play him in his last match, who he's going to finish such a legendary career against.
"So for me, it's an exciting time. We'll see how he comes back but he's going to bring a lot to the game."
The four-time grand slam champion has not played a match since September 2022, taking the whole of this year off and giving birth to daughter Shai in July.
“I am really excited about getting back out on court and competing,” said Osaka. “I always love starting my season in Brisbane and can’t wait to return. The Brisbane International is such a great tournament and will set me up for a brilliant comeback this summer.”
The 26-year-old had played an increasingly sparse schedule before her maternity leave, stepping away from the court several times to prioritise her mental health.
But, speaking to ESPN during the US Open, she said of her time away: “It really raised my love for the sport. It made me realise I’m not going to play forever. I have to embrace the times. I’ve been playing tennis since I was three.
“I don’t think I can predict what I’ll do. I never am able to do that, but it definitely made me appreciate a lot of things that I took for granted.”
Osaka will play her first grand slam since the US Open in 2022 at the Australian Open, which she won in 2019 and 2021, later in January.
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Also playing at the Brisbane International, which takes place from December 31 to January 7, will be Andy Murray.
“It’s great to see the Brisbane International back and I am really looking forward to returning to Queensland this summer,” said Murray.
“I have great memories of winning the title in 2012 and 2013 and I would love to lift the trophy again for a third time in Brisbane.”
Playing her first singles match of 2020, French Open champion Barty was toppled 6-4 7-6 (7-4) by American qualifier Jennifer Brady in the last 16 on Thursday.
All eyes were on Barty with the Australian Open on the horizon, however, she was surprisingly dispatched in one hour, 31 minutes on Patrick Rafter Arena.
Expectations are high as Barty enters her home grand slam – which gets underway in Melbourne on January 20 – as the WTA's top-ranked player but the 23-year-old played down the pressure.
"Oh, I think that's all come you from you guys [media], if I'm being completely honest," Australian star Barty said of the pressure.
"It doesn't change the way that I practice. It doesn't change the way that my team and I prepare. It doesn't change me as a person.
"I think that you guys enjoy the fact that you get something to write about and you have something to talk about.
"Hopefully I make it interesting for you, but I don't look at it beyond that."
It was a difficult day for Barty, who struck 25 winners but 16 unforced errors and could not manage a break point on Brady's serve throughout.
"Of course I would love to go further. And I think, in my point of view, there's no more expectation from however many thousand people were out there today to when I play anywhere else around the world," said Barty as she turns her attention to next week's Adelaide International.
"I think I'm just trying to do the best that I can and obviously try and put my best foot forward and try and play my best tennis, because I know that they have obviously come to see quality tennis regardless of who is on the court.
"They have taken the time out of their day to come and watch some quality tennis. And I know that Jen and I played a quality match today, regardless of the result. And I think now I look forward to kind of doubles here and then maybe going to Adelaide."
The 37-year-old Spaniard, who had surgery on the psoas tendon in his left hip in June, was making his comeback at the Brisbane tournament.
Nadal required medical treatment during his quarter-final loss to Jordan Thompson, and the 22-time grand slam champion said in a post on X: “During my last match in Brisbane I had a small problem on a muscle that as you know made me worried.
“Once I got to Melbourne I have had the chance to make an MRI and I have micro tear on a muscle, not in the same part where I had the injury and that’s good news.
“Right now I am not ready to compete at the maximum level of exigence in five sets matches. I’m flying back to Spain to see my doctor, get some treatment and rest.”
He added: “I have worked very hard during the year for this comeback and as I always mentioned my goal is to be at my best level in three months.
“Within the sad news for me for not being able to play in front of the amazing Melbourne crowds, this is not very bad news and we all remain positive with the evolution for the season.
“I really wanted to play here in Australia and I have had the chance to play a few matches that made me very happy and positive. Thanks all for the support and see you soon!”
Williams and Wozniacki, playing as a pair for the first time, beat the top seeds in the quarter-finals and saved five set points in their opening set against Kirsten Flipkens and Alison Van Uytvanck before sealing a 7-6 (11-9) 6-2 victory to advance to the final.
They will play the winners of Saturday's all-American clash that sees Coco Gauff and Caty McNally take on Asia Muhammad and Taylor Townsend.
Earlier on Friday, Williams came from a break down in the second set to beat Laura Siegemund 6-4 6-3 and book an exciting semi-final against Amanda Anisimova, a 6-2 3-6 6-4 winner against wildcard Eugenie Bouchard.
Asked if she felt in good shape ahead of the Australian Open, Williams said: "I definitely feel that, and I feel like my match yesterday [against Christina McHale] was great.
"It's great that I went three sets, and it's great that I ran so much. I came out today just overall feeling better and doing better.
"Especially the match fitness – I'm super fit, but match fitness is a little different, so I think all these matches are really helping."
Wozniacki will face Jessica Pegula, who progressed when Alize Cornet retired at 6-0 3-2 down, after a routine 6-1 6-4 victory over defending champion Julia Goerges.
Osaka can improve her seeding ahead of the Australian Open if she makes the final at the Brisbane International, and standing in her way is her direct rival in world number two Pliskova.
Two-time major winner Osaka – ranked fourth in the world – had to fend off five break points to close out a 6-3 3-6 6-3 triumph over Kiki Bertens in the quarter-finals.
Reigning champion Pliskova lies in wait in the last four, though she had to stave off two set points in the first before winning 7-6 (8-6) 6-3 against Alison Riske. Petra Kvitova and Madison Keys will contest the other semi-final.
Garbine Muguruza was unable to kick off 2020 with a final appearance as she lost 6-4 6-3 to Ekaterina Alexandrova in the semi-finals of the Shenzhen Open.
Elena Rybakina, who beat Kristyna Pliskova 6-2 7-5, stands between Alexandrova and a first singles title on the WTA Tour.
Not since winning the 2017 Australian Open had Williams clinched silverware – the 23-time grand slam champion a runner-up at Wimbledon (2018 and 2019) and the US Open (2018 and 2019), while she retired in the 2019 Rogers Cup final.
However, Williams ended her drought by defeating fellow American Jessica Pegula 6-3 6-4 in the women's final in New Zealand.
"It feels good. It's been a long time. I think you could see the relief on my face," Williams said during her on-court interview before announcing she will donate her prize money (and a dress from each of her matches) to the bushfire relief effort in Australia.
Williams was almost flawless against Pegula, stepping up her preparations for the upcoming Australian Open with two aces and just one break of serve to capture her 73rd WTA crown – a record stretching across four decades after winning her first in 1999.
After a sluggish start, which saw the top seed fall 3-1 behind in the opening set, Williams eventually found her range en route to the 38-year-old's first trophy as a mother.
After snapping a sequence of five defeats in finals since her 2017 triumph at the Australian Open, Williams will look to equal Margaret Court's record of 24 slam titles at Melbourne Park.
Meanwhile, defending champion and second seed Karolina Pliskova won the Brisbane International for a second straight year after outlasting Madison Keys in three sets.
Pliskova prevailed 6-4 4-6 7-5 over eighth seed Keys for her third victory at the WTA Premier event in Brisbane.
The 23-time grand slam champion had taken a break after losing to Bianca Andreescu at Flushing Meadows.
Having warmed up with a doubles win alongside Caroline Wozniacki on Monday, Williams was back in singles action on Tuesday and cruised past Giorgi 6-3 6-2 in a little over another hour.
"I thought it was solid, technically my second match of the year," she said.
"Spent a long time off from tournament play, match play, so solid is pretty good. It's a great start to build on.
"It's definitely not easy [to get the rhythm back]. It's always hard to play a match early in the season."
Williams will face Christina McHale next and could meet 15-year-old Coco Gauff for the first time in the quarter-finals.
Her doubles partner Wozniacki, who will retire after the Australian Open, also progressed having dropped just one game against Paige Mary Hourigan.
Former world number one and defending Australian Open champion Naomi Osaka had more difficulties in beating Maria Sakkari 6-2 6-7 (4-7) 6-3 at the Brisbane International.
Sakkari had the momentum in the decider when she forged an early break, yet Osaka fought back and converted her second match point after double faulting with her first.
Petra Kvitova and Kiki Bertens, who, like Osaka, started the new season ranked in the world's top 10, also advanced in three sets with respective wins against Anastasia Pavlyuchenkova and Dayana Yastremska, while Jennifer Brady came from a set down to oust Maria Sharapova.
Belinda Bencic was eliminated at the Shenzhen Open, though, the world number eight and top seed beaten 3-6 6-3 6-3 by Anna Blinkova.
Defending champion and second seed Aryna Sabalenka was also knocked out as she suffered a straight-sets defeat to Kristyna Pliskova.
Williams needed just 44 minutes to crush 18-year-old Amanda Anisimova 6-1 6-1 in their last four clash at the WTA International tournament in New Zealand.
The 23-time grand slam champion mixed 17 winners with six unforced errors and lost just nine points on serve. Williams served five aces and converted all five of her break points in a dominant display in the lead up to the year's first grand slam in Melbourne.
The American moved into her 98th WTA Tour final and will be aiming for a 73rd title but first since the 2017 Australian Open when she faces Jessica Pegula.
Pegula upset fifth seed Caroline Wozniacki 3-6 6-4 6-0 in their semi-final.
A two-time runner-up, Wozniacki – who will retire after the 2020 Australian Open – won just 10 points in the third set.
Wozniacki will contest the doubles final with Williams against Taylor Townsend and Asia Muhammad.
In Brisbane, Pliskova lost a first set on a tie-break in which she squandered two set points and saved four, and was staring down the barrel after Osaka broke for a 6-5 lead in the second.
However, Osaka's first serve on match point was overly tentative and a series of errors handed Pliskova the break back. The second seed took full advantage, winning the subsequent tie-break and taking the decider in comfortable fashion to defeat the reigning Australian Open champion 6-7 (10-12) 7-6 (7-3) 6-2.
Earlier, Madison Keys rallied past Petra Kvitova to clinch a 3-6 6-2 6-3 victory, meaning there will be no all-Czech final in Queensland.
At the Shenzhen Open, Russian Ekaterina Alexandrova clinched her maiden WTA Tour title with a 6-2 6-4 win over Elena Rybakina in the final.
