The armband is still up for grabs at Old Trafford ahead of the upcoming Premier League season, following the departure of skipper Nemanja Vidic to Inter.
Manager Louis van Gaal is expected to decide between Rooney and Robin van Persie, though the Dutchman gave nothing away during the club's pre-season tour of the United States.
Darren Fletcher captained the side on three occasions in the USA, with Tom Cleverley deputising against Roma.
Rooney, however, was given the nod to lead in the International Champions Cup final on Monday, in which he scored as United overcame arch rivals Liverpool 3-1.
And Schmeichel - a five-time Premier League champion and UEFA Champions League winner with United - told Sky Sports the England international was primed for the role.
"What I really like about Rooney is his attitude towards playing for United," said Schmeichel.
"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."
United have one more friendly fixture remaining against Valencia at Old Trafford on August 12, before hosting Swansea City in their Premier League opener four days later.
