The Turkish Super Lig broadcaster have reported that a deal is now in place and that the transfer will go ahead.
West Ham reached an agreement with Tore last month a source close to the club told Turkish-football.com but had yet to agree a transfer fee with the Black Eagles.
The Hammers initially reportedly offered £12 million for Tore but the Istanbul giants held out for a higher price.
CNN Turk claim that the Black Eagles wanted £17 million but have agreed to a £13 million fee.
West Ham boss Slaven Bilic is a keen admirer of Tore and is credited with developing his talent during his two-year stint in charge of Besiktas.
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.”
Bilic also confirmed that he tried to bring in Tore last summer but the deal stalled following Besiktas’ reluctance to part ways with their star man.
Tore missed out on the Euro 2016 squad call-up for Turkey due to being suspended for the first two group games so his fitness levels will not be affected by the summer youtnamt