Per the source, the Lions are making a double swoop for the out-of-favour midfielder and Antonio Valencia.
The Lions want Schweinsteiger on loan with a buyout clause option. The 32-year-old still has another two years remaining on his United contract and the Sun claim he would be set to earn £14 million should he stay.
Galatasaray are prepared to pay £2.97 million as a loan fee to United and cover part of his wages next season.
Schweinsteiger has been banished to the reserve side and was not included in the Red Devils pre-season squad according to the Guardian.
The 32-year-old has not played in the pre-season friendlies and Jose Mourinho has made it clear that Schweinsteiger does not feature in his plans next season.
Galatasaray do however, face competition from Istanbul rivals Besiktas according to the Express. A source close to Besiktas did however, tell Turkish-football.com that the club a move looks unlikely considering his current wages.