Yilmaz moved to Glasgow from Turkish top flight club Besiktas in the summer, but has struggled to make an impact and push Borna Barisic out of the left-back slot.
He has featured in a total of seven games so far for the Gers and is currently recovering from a hamstring injury.
Besiktas have been strongly linked with wanting to re-sign Yilmaz, potentially on loan, but there is other Turkish interest in the player.
Galatasaray are one of the teams keen on the left-back and they have been in touch with his agent, according to Turkish outlet En Son Haber.
It is suggested that they have received a positive response from Yilmaz’s agent, as they eye a move for him.
Okan Buruk’s side want Yilmaz initially on a loan deal with the option to make the move permanent at the end of the season.
A bid is now expected to be sent to Rangers soon.
It is unclear what plans new Rangers manager Michael Beale has for Yilmaz, heading into the January transfer window.
