The report claims that the Hammers have been keeping close tabs on the Canadian international and that the East London outfit have made an enquiry.
West Ham are already reported to have reached an agreement with Besiktas over Gokhan Tore and are also interested in Oguzhan Ozyakup.
Hammers boss Slaven Bilic coached Besiktas for two seasons before taking over West Ham last summer so is familiar with his former players.
Atiba has one-year remaining on his current deal but has established himself as one of the Istanbul giants key players.
The 33-year-old forms part of the backbone of the Besiktas side that won the league title last season and will keep his place in the team next season if he were to stay.
The hard working central midfielder had two goals and two assists in 43 appearances for the Black Eagles last season.