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.

Appearing at The Times Cheltenham Literature Festival, she said: "However you look at it, he is me."
Rendell, made a peer in 1997, said she gave Wexford many of her habits as well as her hatred of corporal punishment and a memory for birthdays.
She also paid tribute to the late actor George Baker who played Wexford on TV series The Ruth Rendell Mysteries.
The 80-year-old, from West Lavington, Wiltshire, died of pneumonia on Friday following a recent stroke.
"Where would Wexford have been without George?" Rendell said.
The actor had "made the popularity and the character - a lot of it was George," she added.
"It was a marvellous achievement as an actor to make him more and better than the author intended," she told the festival.
The Ruth Rendell Mysteries ran for 12 series in the 1980s and 1990s on ITV.
She said that, like her, Wexford liked to walk around in London, read poetry and was tolerant - "but not of cruelty or indifference or callousness".
Rendell's latest novel, The Vault, sees Wexford come out of retirement to try to crack an unsolved crime.
Speaking to the BBC, Baker's daughter, Ellie, said her father "absolutely loved Wexford and he loved being Wexford... and he loved the whole thing. It was a joy to him."
His third wife, Louie Ramsey, who died earlier this year, played his wife Dora in The Ruth Rendell Mysteries.
German World Cup winner Bastian Schweinsteiger's future at Manchester United is in doubt following Jose Mourinho's appointment as manager.
[NEXT_CONCEPT]
Author Ruth Rendell has admitted that her famous creation, Ch Insp Reginald Wexford, is based on her.