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.
Third seed Mertens dropped serve in the opening game but instantly recovered and went on to claim a 6-3 6-3 victory over Tsurenko, who was playing her first match back from four months out with an elbow injury.
Next up for the Belgian will be wildcard Wang Xiyu, who came from a set down to beat Sorana Cirstea 3-6 6-1 6-1.
Seeds Ekaterina Alexandrova (5) and Zhang Shuai (8) advanced in straight sets, with Kateryna Bondarenko and Wang Yafan the other winners.
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.
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.
