Schweinsteiger turned 32 on Monday, the same day his locker in United's first-team dressing room was cleared out and he was ordered to train with the reserves by new United boss Jose Mourinho.
He is one of nine players told they are free to go on loan by Mourinho and are not in the United boss's plans.
Schweinsteiger announced his international retirement last week after 12 years, 24 goals and 120 appearances in the famous white shirt of Germany.
At the 2014 World Cup in Brazil he enhanced his reputation as his battling display in Germany's midfield helped beat Argentina in the final.
After winning eight Bundesliga titles with Bayern, as well as the German Cup seven times and the 2013 Champions League, he moved to Old Trafford in 2015.
But his star has fallen after an injury-ravaged season in Manchester which yielded only 31 appearances, and Bayern chairman Karl-Heinz Rummenigge says the door is open for him to return.
"Should he speak at some point about a desire to return, then we will discuss it with him," Rummenigge told magazine Sport Bild.
The imminent return of France's Paul Pogba to Old Trafford, and his own demotion, means Schweinsteiger's days in Manchester look numbered with two years left on his United contract.
"No respect!" tweeted his brother Tobias Schweinsteiger, a former second division player and youth team coach at Bayern.
His reported annual wage bill of Â£7.5 million ($9.9m, 8.8m euros) might put off any potential employer.
The defensive midfielder has been linked to AC Milan and Inter Milan, as well as Paris Saint-Germain, while any Major League Soccer club would be delighted by the profile boost his arrival in the USA would bring.
And Schweinsteiger is too young to retire, according to his ex-Germany and Bayern team-mate Oliver Kahn.
"I'm sure that with all his experience, Basti can still be the leader of a top team. He is definitely too young and too strong to retire," Kahn wrote on his Facebook page.
Schweinsteiger, who married tennis player Ana Ivanovic last month, is weighing up his options.
Rummenigge says he does not expect Schweinsteiger to struggle to find a new employer and it seems he has a future with Bayern when he hangs up his boots for good.
"It's important to bind players who deserve it to the club," Bayern captain Philipp Lahm told magazine Sport Bild.
"Basti knows everyone and everything here, he would be predestined to work in the team."
Bayern have a history of finding jobs for their chosen sons and Rummenigge has already said they will "discuss the matter in due course".