The report states West Ham have made a bid for the Turkish international and that talks with then Hammers will continue though-out the week.
Yilmaz was let on the bench during Galatasaray’s 2-2 draw with Sivasspor on Saturday. The 29-year-old came on in the second half to score his sides first goal.
West Ham have reportedly offered Yilmaz an offer worth €4 million per-season on a three-year deal according to Futbol Arena.
The 29-year-old striker still has four-years left on his contract and is the first-choice striker at Galatasaray who won the league and cup double last season.
The Lions also booked a place in the Champions League for the fourth consecutive season.
The Istanbul giants take on Osmanlispor in their second game of the 2015-16 Super Lig season this weekend.