Morgan is expected to step down as captain on Tuesday, with a news conference at Lord's having been arranged.
It is thought Buttler, the vice-captain, will replace Morgan, who has been in charge since 2014. He has led England to World Cup glory, as well as the T20 World Cup final.
The Ben Stokes-Brendon McCullum red-ball tenure started with a 3-0 series win over New Zealand, but while new white-ball coach Matthew Mott saw his team claim an easy series victory in the Netherlands earlier in June, Morgan is now set to quit his post and retire from international cricket, having passed 50 just once in his last eight ODI outings.
Vaughan believes England will forever be indebted to Morgan, writing in The Telegraph: "There have been many Test captains who have made an impact on the history of English cricket during their time in charge, but there has only been one white-ball captain that has done so – Eoin Morgan.
"The freedom and fearless approach that he's given this white-ball team is going to be with England forever. He's going to be remembered forever and can now sit back and be very proud of what he's achieved as an individual leader.
"English cricket is in an exciting place – you've got this white-ball group of players that is so deep and so full of power, and the question is how many are going to be left out that should be in the side. And a lot of that is down to what Morgan has put in place."
And Vaughan feels Buttler, fresh off some wonderful displays against the Netherlands and in the Indian Premier League, is the perfect replacement.
"For me it's a no-brainer that Jos Buttler takes over that role. He's the best white-ball player in the world, he's got a very smart cricket brain, and he's got that calmness you need," Vaughan wrote.
"I guess his personality might be different from Eoin. The one thing that Jos will have to be very, very good at is staying the same when he doesn't have a good game or two. That has been Eoin's massive strength – he has never changed and even last week in Holland after getting two noughts, I bet he was still the same person in the dressing room."
Vaughan also believes Buttler could provide the solution to a major weakness in England's Test side.
"That might not be all Jos could do for England, though. Kumar Sangakkara said something this week, which I thought was ridiculous the first 10 minutes I thought about it: Buttler should be England's Test match opener," Vaughan continued.
"And then it hit me that with this Test match team and the way that they're playing: this might be an idea worth exploring. England have this fearless, aggressive nature. If something as radical Buttler as Test opener was ever going to work it would be now, under this management group of Ben Stokes and Brendon McCullum.
"I wouldn't say it's a sensible option – because it's not sensible. But I don't think some of the decisions that this Test match team are going to be making are going to be sensible.
"What's the most aggressive, radical thing that we could think of? Let's go. Jos opening in Test cricket is quite radical. Just go for it."
Buttler has not featured for England in a Test since the 2021-22 Ashes in Australia, where he failed to impress with the bat, top-scoring with a 39 in Brisbane.
England were held to consecutive draws across Tests in Antigua and Barbados before faltering in the winner-takes-all decider in Grenada.
Root's tourists recovered from 90-8 to 204 in the first innings, but were never ahead in the game as Kraigg Braithwaite's side mustered 297 before again bowling England out for just 120.
That left Brathwaite and opening partner John Campbell to secure a 10-wicket victory as they required just 4.5 overs to chase down 28 on Sunday, condemning England to a fourth consecutive series defeat.
England have won just one of their last 17 Tests and are winless in their last nine red-ball outings, their longest such streak in the format since a run of 10 between August 2013 and July 2014.
Questions over Root's captaincy remain prominent with the ECB searching for a new managing director and coach, and Vaughan believes his fellow Yorkshireman needs to take some time to consider his future.
"Let Joe Root sleep on it for a week or so," Vaughan said to BT Sport of Root's future at the helm.
"I fear this red-ball team might go further backwards before it goes forward and you're going to have to have a lot of energy as a leader, a captain you're going to have to have a huge amount of energy to wake up every morning to captain this side.
"Generally in English conditions, the Test match team win lots of games, win lots of series – well last summer they lost to New Zealand and they were losing to India, so I don't see this Test match side suddenly becoming a team that consistently wins series after series and that's in English conditions.
"So Joe is going to have to find a huge amount of energy and he's also going to have to improve, because tactically in this game he was a long way short. The England side fell a long way short.
"I don't see too many players, out of this England Test team, who can suddenly come in and spark England into getting 450 consistently against the better opposition, when the ball is moving about.
"That's why I do think there could be some darker days ahead and it's going to take a leader with a lot of energy to try and get this Test match team right."
Paul Collingwood took temporary charge for the series in the Caribbean after Chris Silverwood was dismissed following Ashes disappointment, but the identity of England's next permanent coach remains unclear.
Vaughan would like to see England appoint former Australian coach Justin Langer, who guided his country to T20 World Cup success at the end of last year before lifting the Ashes.
"I would personally go for Justin Langer – he's the sort of leader that England need at the moment," he added.
"Then it is a conversation with Joe Root to see if he's still got the energy to take England forward and even then I'd debate it.
"You could give it to Ben Stokes to the end of next year's Ashes and then hope that a younger player like Zak Crawley is ready."
England's dismal 10-wicket defeat to West Indies in the third Test in Grenada meant a 1-0 series reverse, coming off the back of a 4-0 Ashes thrashing at the turn of the year.
Those are two of five consecutive Test series defeats for England, who had won four in a row before then. They have not endured a worse such run since six without a win between 1987 and 1990.
England are also winless in nine Test matches, their worst sequence since a barren stretch of 10 between 2013 and 2014.
Alastair Cook survived that spell as skipper, but Root – the only man to captain England in more Tests (64 for Root, 59 for Cook) – is now under intense pressure.
And Vaughan, fourth on the list of matches as England captain (51), suggests the time has come for one of the world's best batsmen to focus solely on his own game.
"He's taken it as far as he possibly can," Vaughan told BBC Radio 5 Live.
"If he rings me in the next week and asks for some advice, I'll be dead honest: I'd tell him to step down.
"Will England be any worse off not having him as a captain? I don't think they would, because they are going to get his runs and a senior player.
"They'll get a great role model – I don't think there is a better role model in English cricket."
Root has averaged 46.4 with the bat as captain, down from 52.8 up to that point, with all 53 prior matches coming under Cook.
Only in 2013 (34.5) has his batting average been lower across a calendar year than his 35.8 so far in 2022.
Vaughan was charged with bringing the game into disrepute for allegedly saying there were "too many of you lot", referring to Asian players prior to Yorkshire's T20 match against Nottinghamshire in 2009.
Christopher Stoner KC, representing Vaughan, claimed his client was denied "due process" during the ECB's investigations into the allegations.
Stoner also labelled the investigation "wholly and woefully inadequate" as the Cricket Discipline Commission (CDC) hearings concluded with closing submissions in London on Tuesday.
"The investigation was wholly inadequate," Stoner said. "Due process matters and is the cornerstone of law.
"In our submission it was sent on holiday by the ECB. It raises a real question of fairness [of this investigation]. Mr Vaughan has not been accorded fairness."
ECB lead counsel Jane Mulcahy focused on a series of historic tweets made by Vaughan in her closing submission.
"If a person has a tendency to make racist comments, they have a tendency to make racist comments," Mulcahy said.
"Although Michael Vaughan now purports to be a changed character, Vaughan in 2009 was the same person who shortly afterwards (in 2010) sent two tweets complaining about foreigners… [he] still held the same 'unacceptable' views seven years later when he sent further tweets concerning Muslims and potential terrorism… the supposedly lighthearted but offensive expression in the tweets is very similar in tone to the comment made on 22 June 2009."
Stoner pointed to the testimony of Ajmal Shahzad, one of the four players Vaughan was alleged to have made the comment about, who said that Vaughan "wasn’t that way inclined" to making racist comments as important counter evidence.
Vaughan's defence team went on the offensive with a 32-page closing written submission and a 22-page storyboard of Sky's footage of the pre-game huddle from that day.
Stoner said of the footage: "[It is] Inherently improbable that such serious and unacceptable words were spoken to team-mates just as a game was starting, in the presence of a cameraman and almost certainly a microphone."
Mulcahy also defended the ECB's investigation from Stoner's claims of inadequacy, labelling it an "extraordinary amount of bitter and inaccurate correspondence".
