The Red Devils sacked Louis van Gaal on Monday night despite the Dutchman ending their three-year trophy drought at the weekend thanks to a 2-1 extra-time victory over Crystal Palace at Wembley.
Mourinho is the clear favourite to succeed the 64-year-old at Old Trafford after being heavily linked with Van Gaal’s job since he received the chop at Stamford Bridge in December.
The former Inter Milan and Real Madrid manager is thought to be on the brink of taking over the reins at Old Trafford as United look to make a return to the Premier League’s top four.
United midfielder Schweinsteiger, who moved to Old Trafford from Bayern Munich in a £6.7m deal last summer, reacted to reports that Mourinho will become the new Red Devils boss.
“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,” Schweinsteiger told the media at a Germany press conference on Wednesday afternoon.
The Red Devils finished in fifth place in the Premier League table.
MORE: Man United latest news
MORE: Arsenal latest news
MORE: Chelsea latest news
MORE: Liverpool latest news