The report claims that the Hammers had a £12 million bid turned down over the summer but that Galatasaray could be prepared to sell Yılmaz over the winter transfer window for the right price.
Yılmaz is reported to be unhappy at Galatasaray and Hammers boss Bilic has openly stated in the past that he is a fan of the Turkish international.
“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.
Yılmaz was closely linked with a move to Italy and England prior to joining Galatasaray. The experienced striker scored 58 goals in 93 games for the Black Sea Storm before making his move.
The Galatasaray striker has won the league title and the Turkish Cup twice since joining the club. Yılmaz has three-years left on his contract and is paid in the region of £2.5 million per-season.