The Turkish international was one of Slaven Bilic’s key signings during his spell as Vodafone Arena boss – arriving from Rubin Kazan during summer 2013, initially on loan.
The 24-year-old has gone on to bag 19 goals and 26 assists in 111 appearances for the Black Eagles, registering 24 league outings last season as Besiktas claimed the Super Lig title.
Bilic is keen to reunite with the one-time Chelsea youngster at the Olympic Stadium and has already tabled a loan bid to sign Tore for next season.
However, Besiktas are only interested in a permanent transfer and want £12million – but talks between the two clubs are still ongoing.
Tore’s arrival would continue a summer of impressive recruitment from the East Londoners, having already signed Valencia winger Sofiane Feghouli and Borussia Monchengladbach’s Havard Nordtveit.