The report states that Besiktas want €7 million for the Canadian international but that the Hammers do not want to go over the €5 million mark.
Hutchinson was a key player in the Black Eagles side under the stewardship of Slaven Bilic last season.
Bilic coached the Istanbul giants for two seasons and guided Besiktas into the Europa League quarter-finals knocking Liverpool out of the competition along the way.
Besiktas are however, facing a crisis in midfield with central midfielders Veli Kavlak and Tolgay Arslan both out injured until the turn of the year.
The Eagles would be left lacking depth if Hutchinson were to leave and it would force the club into signing another midfielder before the end of the transfer window.
The former PSV player provided three assists and two goals in 43 appearances for the Black Eagles last season.