Burak Yilmaz has handed in a transfer request in order to force a move to West Ham, The Times report.
The Galatasaray striker has been heavily linked with a switch to the Premier League side since the weekend.
Turkish newspaper Hurriyet reported yesterday that Hammers vice-chairman Karren Brady is to fly to Turkey to negotiate.
Yilmaz, 30, has a contract until 2019 and Galatasaray are under no pressure to sell, but his transfer request could change their stance.
He has nine goals in 15 Super Lig games this season, and is also aiming to play a part at Euro 2016, with 19 goals in 42 appearances for his country.
West Ham are also negotiating to sign Fenerbahce forward Emmanuel Emenike, with SkySports reporting rival interest from Crystal Palace.