The Sun meanwhile reported that Tore will wear the number 7 shirt for the Hammers next season.
West Ham have been keeping close tabs on Tore for months and made a bid for him last summer which fell through.
Hammers boss Slaven Bilic is credited with getting Tore back on track after a few shaky seasons in Germany and Russia.
Bilic transformed Tore into one of the most sought after players in Turkey and a regular Turkish international call-up.
The Croatian manager revealed that he tried to sign Beşiktaş winger Töre at the start of the season in an interview with the Socrates magazine.
“Gökhan is a legend he is a top guy and a great player of course I would love to have him in the team,” Bilic said.
“I tried to sign him at the start of the season but it was not the right time.”
Tore’s agent Ahmet Bulut recently advised his client to consider a move to Europe at the end of the season.
Bulut stated that Tore should make the move if he wants to become the next Arda Turan and confirmed that he has received offers.
“I can’t say if Gökhan will stay or league but if he wants to forge a career in Europe this summer would be ideal.
“I can’t say where he wants to go but if he wants to be like Arda he needs to consider it.”
The Turkish international winger is no stranger to England having first made a name for himself in the Chelsea youth setup.
The explosive winger had six goals and nine assists in all competitions for the Super League champions this season.