Per the source, the Lions are plotting a swoop for the out-of-favor midfielder who has been deemed surplus to requirements by head coach Jose Mourinho.
Galatasaray are keen on strengthening their midfielder with an experienced player making the former World Cup and Champions League winner an ideal choice.
Schweinsteiger has already been banished to train with the reserves and was left out of the pre-season as well as Community Shield squads.
Mourinho has made it clear that the 32-year-old does not have a future at the club and following the arrival of Juventus star Paul Pogba he will be almost certain to part ways with United.
Schweinsteiger has also been linked with Super Lig rivals Besiktas as well as a move to Germany and the Serie A.
Galatasaray also have their sights set on Liverpool midfielder Lucas Leiva and sporting director Levent Nazifoglu has given head coach Jurgen Klopp an ultimatum that he must make a decision by tonight regarding the players future.
Nazifoglu was quoted as saying: “We have asked for a response regarding Lucas Leiva by tonight. If we don’t receive one, we’ll pursue alternative players.”