Mourinho has been strongly linked with the Old Trafford post should, as expected, the current incumbent Louis van Gaal be relieved of his duties at the end of the season.
A number of former United players have recently backed Ryan Giggs for the job, but Sheringham has gone against the trend.
"There's an awful lot of talk about Mourinho going there and I wouldn't be averse to that. He has the belief that he deserves to be Manchester United manager and that's a big part of the United job," he told Squawka
"You have to look at who has played well for Manchester United. Van Gaal's job is to make the players play to the best of their capabilities.
"David De Gea is probably the only one performing to his potential, and that's not good for a football club that your goalkeeper is the only one doing that. It has to mean you are doing something wrong."
Mourinho, 53, has been out of work since he was sacked by Chelsea in December.