The 25-year-old full-back has been placed on the transfer-list by United following the acquisition of Matteo Darmian from Torino.
Galatasaray have been closely linked with the Brazilian international who has also received interest from Fiorentina and Napoli.
Rafael – who was hampered by injury last season – fell out of favour with manager Louis van Gaal who preferred Antonio Valencia at the right-back position.
With the Brazilian full-back going into the final year of his contract at Old Trafford, United are keen to offload Rafael for a cut price.
Rafael provided one assist and scored on goal in 17 appearances for Manchester United last season.
Galatasaray won the league title for a record 20th time last season and booked a place in next seasons Champions League group stages.