The Red Devils are being widely tipped to appoint the former Chelsea boss as their next permanent manager by the end of the week.
United sacked Louis van Gaal on Monday night despite the 20-time English champions ending a three-year wait to win silverware last weekend thanks to a 2-1 extra-time victory over Crystal Palace in the FA Cup final at Wembley.
Mourinho led Chelsea to a fourth Premier League title and secured the League Cup trophy during his second spell at the west London club before he was given the chop following a disastrous start to the 2015/16 season.
Germany midfielder Schweinsteiger is optimistic about United’s prospects under the 53-year-old ahead of his anticipated appointment.
“We don’t know who will be our new manager yet, but it looks like it’s going to be an interesting derby in Manchester next season,” Schweinsteiger told a media conference.
“The enthusiasm for football in Manchester is great and, if Mourinho does take over at United, I think all of the players want to train under him so it’s going to be quite a special season.”
The 31-year-old midfielder moved to United in a £6.7m deal from Bayern last summer. Schweinsteiger scored his only Premier League goal against champions Leicester City.
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