The report goes onto state that the Istanbul Giants have set aside €13 million for the Netherlands international.
Van Persie will enter the final year of his Manchester United contract next season. The 31-year-old is understood to be unhappy with the lack of first team football this season.
Galatasaray could face competition from bitter rivals Fenerbahçe after it Milliyet claimed that the Yellow Canaries have stepped up efforts in their attempt to sign the Dutch star.
The Lions won the league title for the 20th time last season and went on to lift the Turkish Cup for the 16th time to win their sixth double.
Galatasaray will hope the lure of playing regular first team football, a lucrative contract and Champions League action will be enough to persuade Van Persie to join.