Manager Louis van Gaal is still deciding over his skipper for the forthcoming season and dropped the biggest hint yet that it would be Rooney by naming him captain for the 3-1 International Champions Cup win over Liverpool.
It was the first time the England striker led the team on their five-match tour of the USA, while Scotland midfielder Darren Fletcher and Robin van Persie, whom Van Gaal made Netherlands skipper for the World Cup, are also in contention.
Schmeichel told Sky Sports: "What I really like about Rooney is his attitude towards playing for United.
"Wherever the manager tells him to play he will do that with a smile on his face.
"He realises his importance to the team and if he has to play in midfield, it's for the team. You get that with experience and matureness.
"So I think he's the obvious choice for the captaincy."
Much has been made of Van Gaal’s approach to transfers this summer, with United yet to make a first-team signing since the Dutchman took charge following the World Cup.
Van Gaal has insisted every one of United's squad has a chance to prove themselves in pre-season and has suggested he may not need to bring any more players in.
But Schmeichel, who made almost 400 appearances in an eight-year stay at Old Trafford, believes Van Gaal is being clever not to pile pressure on himself and the club.
"He came out in the media saying he might not need players and I think that's clever because by saying he needs four players, for example, it puts himself and the team under more pressure," Schmeichel said.
"I don't think they've ever been under as much pressure as they are for this season so saying 'what I've got is good enough – I can work with this' is clever.
"But I think managers always like to add a little bit to the squad and we've still got a little bit of time to go until the transfer window closes, so I think one or two players will come in."