Rooney led England to their 5-0 win over San Marino on Friday night.
After that game, Roy Hodgson was moved to suggest Rooney would succeed Steven Gerrard as England captain.
And having benefited from working with the 26-year-old at Old Trafford, Cleverley would have no problem if his United team-mate inherited the armband on a full-time basis.
"Wayne has all the attributes to be a good captain," said Cleverley.
"He is a winner. You could tell that before the game because he was quite vocal in the dressing room.
"I am not the manager but I would certainly be happy playing under a captain like Wayne Rooney.
"He is a fantastic player and a good leader."
England are likely to make a number of changes in defence for Tuesday's World Cup qualifier with Poland in Warsaw, with potentially only Phil Jagielka keeping his place from the back four that faced San Marino.
Glen Johnson will almost certainly return from suspension and there is no need to save Joleon Lescott from another booking that will rule him out of England's next competitive game, as that is the return fixture against San Marino next March.
It leaves only the choice about whether to keep faith with Leighton Baines, who has also started all four England games this season, or Ashley Cole, who would be winning his 99th cap and whose talent has never been in dispute even if his off-field behaviour leaves something to be desired.
"We are really lucky to have some great left-backs in this country," said Cleverley.
"Baines and Ashley Cole are both fantastic players.
"They both get forward, are natural defenders and I wouldn't enjoy playing against either of them.
"If you include Ryan Bertrand and Kieran Gibbs as well, we are really lucky with the standard of defenders we have.
"Whoever is selected on Tuesday night, I am certain they will do a good job."
While the debate about Great Britain's female football participation in future Games continues, it has already been confirmed the men's team was a one-off experiment for London 2012 that will not be repeated.
Few will mourn its passing into history, but Cleverley has good reason to be thankful for its existence.
After all, it gave him the leg up required to claim a place in the international arena.
"The Olympics were very important for me," he said.
"It was perfect for me to get some games and play competitive matches on a big stage.
"Roy Hodgson was at a couple of the games, I was in good form and I have managed to carry that on into the season."
Not that Cleverley is taking anything for granted ahead of Tuesday's game.
He might have started all four games for England this season but it is the first time Steven Gerrard and Wayne Rooney have both been available, so even though Frank Lampard is missing with a calf problem, starting berths are at a premium.
"I am really proud to have started all four games so far," Cleverley said.
"I am always happy to represent my country, but I have a bit more hard work and a few more games ahead of me before I become established.
"There are plenty of top class midfielders in the Premier League."
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