The report claims that the Hammers have put forward an offer that will see the Premier League outfit pay €6 million upfront and a further €1 million instalment if the club ensure Europa League football.
Galatasaray could be tempted to accept the offer considering the club is facing the prospect of a UEFA competition ban for failing to meet Financial Fair Play regulations.
Yilmaz does however, still have another three-years remaining on his contract and earns €3 million per-season.
West Ham boss Slaven Bilic has openly stated in the past that he is a fan of the Galatasaray star who he knows well having coached in Turkey for two-seasons during his time in charge of 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.
Yılmaz has lifted the league title as well as the Turkish Cup twice since joining the club and has been one of the most constantly prolific strikers in the league over the past few seasons.