Per the source, the Lions are willing to contribute more than half the Germany international midfielder’s salary next season.
United boss Jose Mourinho has already made it clear he has no plans for Schweinsteiger after banishing him to the reserve side and leaving him out of all the pre-season friendlies.
The 32-year-old was left out of the pre-season squad and the Community Shield side that travelled to Wembley to take on Premier League champions Leicester City.
Schweinsteiger still has another two-years remaining on his contract and would be set to receive close to £14 million if he were to remain at the club.
Galatasaray are currently in talks with Liverpool over the transfer of Lucas Leiva but have also been in negotiations over a loan deal for Schweinsteiger if the deal falls through.
The Lions could face competition from Istanbul rivals for his signature as Besiktas are also in the race to sign the former World Cup winner.
The former Bayern Munich star managed just 13 Premier League starts for United last season and was plagued with injuries.
Galatasaray have had a busy week completing the transfer of Switzerland international Eren Derdiyok, Belgium international Luis Cavanda and HSV midfielder Tolga Cigerci.