Per the source, Galatasaray want to loan the 32-year-old midfielder next season.
Schweinsteiger will almost certainly not play at the club next season after being banished to train with the reserve side.
The 32-year-old was not included in the United pre-season squad and has not featured in any friendlies according to the Guardian.
United faced a barrage of criticism from Bayern Munich, Schweinsteiger’s brother for the way Schweinsteiger has been treated.
A member of world players’ union FifPro Dejan Stefanovic went as far to claim Jose Mourinho should even serve jail time as a result of how he has treated the experienced midfielder.
“It’s clear bullying,” Slovenia FifPro member Dejan Stefanovic told the BBC.
“In Slovenia, we would have indicted Mourinho and asked for the highest penalty – three years in prison.”
United have given Schweinsteiger the green light to move however, the Sun report that he would stand to make £14 million over the duration of his two year contract should he decide to stay.
The Red Devils may consider Galatasaray’s offer if a loan fee or agreement was made over paying some of Schweinsteiger’s wages next season.
The Express meanwhile report that Besiktas are interested in the World Cup winner but a source close to Besiktas told Turkish-football.com that the club a move looks unlikely considering his current wages.
Bayern Munich CEO Karl-Heinz Rummenigge has also said they would be open to discuss the possibility of Schweinsteiger returning to the club.
“Should he [Schweinsteiger] speak in the future at some point about a desire to return, then we will discuss it with him,” Rummenigge told Bild .