Mourinho is reportedly keen to free up his considerable £130,000-a-week wages just in case the United boss is tempted to dip back into the transfer market in January.
"It's great to see him back".
Schweinsteiger, who has previously said United will be his last club in Europe and vowed to be ready for the team if required, wrote on his Twitter account on Monday: "Felt great today!"
Schweinsteiger is unable to play a part in Thursday's Europa League trip to Fenerbahce given he is not registered for the competition.
"I still think he could be valuable to the squad".
Writing in his weekly blog, he said: "I strongly believe that the team is on the right track, after all the hard work we put on the pitch".
After making 18 Premier League appearances last season, 32-year-old Schweinsteiger has not played for United since the Portuguese took over as manager from Louis van Gaal this year.
"It's great to see him back", Shaw said.
Mata scored the all-important goal that dumped rivals Man City out of the EFL Cup and was given the captain's armband for the 0-0 draw with Burnley on Saturday.
Mourinho was already facing a sanction for questioning whether it would be hard for referee Anthony Taylor to have a "very good performance" in a game against Liverpool on October 17, which finished 0-0.
"Obviously we'll keep that between us two".
"They had enough chances to win the game [against Burnley]".
Reports suggested Mourinho sat down with Mata to discuss his future soon after arriving at the Theatre of Dreams, but the Spain global insists no such conversation took place.
He added: "It's a shame we couldn't get the three points on Saturday, but we have to keep playing the way we're playing".