The report states that Jose Mourinho has no plans for the Ecuador international next season and that he will consider offers.
Galatasaray are set to enter negotiations with Valencia’s agent over the next few days.
The 30-year-old still has another year remaining on his contract but United are unlikely to offer an extension.
Valencia played 854 minutes of football last season making 14 appearances and providing three assists in total.
Galatasaray won the Turkish Cup last season but finished the league in 5th place and will not compete in European competition football next season due to a UEFA ban issued for failing to meet FFP regulations.
Valencia has made 245 appearances for United over the past seven seasons during which time he has lifted the Premier League twice.
The experienced winger is currently on international duty with the Ecuador national team competing in Copa America 2016.