Rooney was handed the captain's armband by Roy Hodgson against San Marino on Friday, in the absence of usual skipper Steven Gerrard, who missed the tie through suspension.
The home side cruised to a comfortable 5-0 victory and Cleverley insists he would be more than happy to see his Manchester United team-mate inherit the role on a more permanent basis.
"Wayne has all the attributes to be a good captain," he told Sky Sports.
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"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 face Poland on Tuesday night, with the game posing a number of selection headaches for Hodgson, particularly at left-back.
Leighton Baines has emerged as a serious contender to Ashley Cole for a place in the starting XI, but Cleverley believes the dilemma is one Hodgson will welcome.
"We are really lucky to have some great left-backs in this country," added 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."