The Lion’s are on the hunt for a new defensive midfielder after Felipe Melo signalled that he may be leaving the club.
Melo has a year remaining on his contract meaning the Istanbul based club will need to sell this season if they do not want he to go for free next season.
The report goes onto state that Manchester United are also interested in signing the Netherlands international.
Wesley Sneijder is understood to be involved in negotiations in an attempt to convince his international team mate to join Galatasaray.
De Jong has score three goals in 29 appearances for AC Milan this season.
Galatasaray ended the 2014-15 season with 77 points following their 1-1 draw with Rizespor in their final game of the season.
The Lions have become the first Süper Lig club to wear four stars on their shirt following the championship.
The Istanbul giants will qualify directly for the Champions League group stages next season.