The report claims that the Hammers have made a bid that will see the east London based side pay €6 million upfront and a further €1 million instalment if the club ensure Europa League football.
Yilmaz added further fuel to the transfer speculation when he missed training yesterday.
The Turkish international striker also caused a media storm after he waved to fans following his goal in the 3-2 defeat against Osmanlıspor yesterday.
Yılmaz scored a stunning goal against the Ankara based side but his goal celebration raised the question of whether he was bidding farewell to Galatasaray supporters.
Galatasaray are currently facing up to two years out of UEFA competition football for failing to meet Financial Fair Play regulations and could be tempted to sell in order to comply with the rules.
Yılmaz has been one of the most prolific strikers in the league over recent years and has scored 81 times in 140 games for the Lions.
The 30-year-old striker also has won the league title twice and has years of Champions League experience.