The Dutchman only arrived at St Mary's last summer but his future is unclear after he played just 22 times in the Premier League last term.
And now, according to Fotospor, Gala are lining up a bid for Clasie, although their bid may fall well below what Southampton would require to sell.
It has been suggested that the Turkish club will open the bidding at just £3million, even though Saints paid £8m to land him this time last year.
Galatasaray have been busy trying to land midfield reinforcements all summer but have been forced to turn their attention to Clasie after failing in their bid for Liverpool's Lucas Leiva.