The report claims that the Turkish international star held a meeting with head coach Mustafa Denizli requesting that the club grants his permission to leave.
Yilmaz is reported to have held talks with West Ham and that initial negotiations have gone well.
The Hurriyet claim that the Hammers have made a €7 million offer for Yilmaz and that vice-chairwoman Karren Brady will travel to Turkey to complete the transfer.
The Hammers are however, currently the only side to have confirmed interest in the prolific striker with head coach Slaven Bilic confirming that Yilmaz is on his winter transfer list.
Bilic told Turkish sports newspaper Fanatik: “Burak Yilmaz is one of the players currently on our transfer list.”
Yılmaz dropped a further hint that he could be heading for a Galatasaray exit after he waved to fans following his goal in the 3-2 defeat against Osmanlıspor on the weekend.
The Turkish international defender waved at fans after scoring his stunning goal. The celebration was interpreted as a final farewell to the Lions supporters.