Tore, who built a good relationship with Slaven Bilic when the pair were at Besiktas, has been the subject of a €10 million (roughly £7.4m) offer from the Hammers, with Bilic keen to reunite with his former star.
However, Besiktas officials have responded to the approach, with the report claiming that they are unwilling to sell Tore this month. They have instead told Bilic to revive his interest in the winger in the summer, should he still want to bring the 23-year-old to the Premier League.
Tore, who signed for Chelsea as a highly-rated youngster before leaving in 2011 without a first-team appearance to his name, has scored 17 goals and created 24 more in 96 games for Besiktas.
He has also played for Hamburg, Bayer Leverkusen and Rubin Kazan, and has been capped 26 times for the Turkish international side.
Tore had also been targeted by Bilic last summer, following his arrival at the Boleyn Ground, and it is clear he is a big fan of the winger. If Besiktas hold firm on their stance and refuse to sell him this January, it is likely that West Ham will attempt to sign him again at the end of this season.