The Turkish sport channel state that West Ham manager Slaven Bilic wants to bolster his attacking options with the 29-year-old.
The report goes onto state that the Hammers are preparing a bid for Yilmaz however, are unwilling to match Galatasaray’s €15 million valuation.
Yilmaz extended his contract with Galatasaray until 2019 last season and earns €2.8 million per season plus bonuses.
The Turkish international striker has been in prolific form over the last few seasons and has scored 72 goals in 121 games for the Lions since joining from Trabzonspor in 2012.
Bilic is reported to be a keen admirer of Yılmaz from his time managing Beşiktaş.
“Burak Yılmaz is a player who has scored over 60 goals in his last two seasons of course I think he is a very good striker,” Bilic was quoted as saying on Lig Tv.
The 29-year-old striker scored eight goals in nine Champions League appearances in the 2012-13 season, playing a crucial role in guiding Galatasaray to the quarter-finals.