The report also claims that the Hammers have reached a £13 million transfer agreement to sign the Besiktas winger.
A source close to Besiktas told Turkish-football.com that a deal was imminent and that talks have been ongoing all week.
West Ham initially offered £12 million but Besiktas were holding out for a higher transfer fee.
Tore established himself as one of the most highly sought after players in the Turkish Super Lig under the stewardship of Slaven Bilic who coached Besiktas for two seasons before taking over at West Ham.
The Turkish international winger is no stranger to Premier League football having first risen through the ranks of the Chelsea academy before moving to HSV Hamburg and then Rubin Kazan before his transfer to Besiktas.
Tore had six goals and nine assists in all competitions for the the Turkish champions last season.