The Hammers had already agreed personal terms with Tore but were locked in talks all week over his transfer fee.
West Ham initially made a £12 million bid but Besiktas were holding out for a higher fee. The Black Eagles wanted £17 million but have agreed to a £13 million offer according to CNN Turk.
The transfer is expected to be officially unveiled on Wednesday 1 June 2016 and will mark the Istanbul giants first sale of the season.
Tore will become the first player to join Slaven Bilic from his former side – Besiktas – since taking over at West Ham last summer.
Bilic got the best out of the explosive winger during his time in charge of the Eagles and established a close bond.
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.”
The Hammers made a bid for Tore last summer as well but the deal stalled. The East London based club kept tabs on the Turkish international all season and Besiktas had already prepared for his replacement in January bringing in Aras Ozbiliz from Spartak Moscow.