Mourinho has already made three big summer signings and is closing in on Juventus midfielder Paul Pogba.
Schweinsteiger, 31, looks like being one of nine players who may leave.
The midfielder, who has announced his international retirement, trained on his own as United's full squad trained together for the first time.
Forwards Adnan Januzaj, Will Keane and James Wilson, midfielder Andreas Pereira and defenders Cameron Borthwick-Jackson, Tim Fosu-Mensah, Tyler Blackett and Paddy McNair are also reportedly set to be made available, though some may only leave on loan.
With all the other senior players who were absent from the recent tour of China present, Schweinsteiger's absence from Thursday's squad training session is being viewed as significant.
United paid a reported £14.4m to sign him from Bayern Munich 12 months ago but the move did not work out as planned.
Injury restricted Schweinsteiger to 31 appearances and he was involved in just seven of United's final 34 games, including missing the FA Cup final win over Crystal Palace.
He injured a knee on international duty in March and only returned at Euro 2016, when he scored on his first appearance, as a substitute, against Ukraine.
Of the other players who did not train with the United squad on Thursday, Januzaj has had the biggest fall from grace.
The 21-year-old Belgian was one of the star performers in David Moyes' single season in charge, when he earned himself a big increase in a five-year contract.
But his progress was not maintained and he was loaned to Borussia Dortmund last season.
He did not start a league game and the German side were happy to let United cut the deal short midway through the season when Louis van Gaal's squad were badly affected by injury.
He then featured for 17 minutes in three substitute appearances.
Since his arrival, Mourinho has signed Villarreal defender Eric Bailly for about £30m, midfielder Henrikh Mkhitaryan from Borussia Dortmund for an undisclosed fee and former Paris St Germain striker Zlatan Ibrahimovic on a free transfer.
German World Cup winner Bastian Schweinsteiger's future at Manchester United is in doubt following Jose Mourinho's appointment as manager.