Negotiations are at an advanced stage and the Black Eagles are prepared to part ways with Tore for the right price.
Besiktas want £12 million for the Turkish international but the Hammers have been trying to iron out a better deal.
The Istanbul giants planned for Tore’s departure in January after bringing in Araz Ozbiliz from Spartak Moscow.
Ozbiliz will join up and coming winger Kerim Frei as well as Ricardo Quaresma and Olcay Sahan on the flanks.
Besiktas boss Senol Gunes started preparing for life without Tore towards the end of the season when he was dropped for Frei and Quaresma.
Tore does have experience playing in England. He joined the Chelsea academy aged 16 and rose through the youth ranks before joining Bundesliga outfit HSV Hamburg.
West Ham boss Slaven Bilic has made no secret of the fact that he wants Tore to join him in East London.
The Croatian manager got the best out of Tore during his time in charge of Besiktas and established a close relationship with the explosive winger.
Speaking to Turkish sports newspaper Fanatik in a recent interview Bilic said: “Everyone knows how much I like Tore he is an excellent player and I want him at West Ham.”