The Hammers have been closely linked with Tore all season and head coach Slaven Bilic has confirmed that the Turkish international is one of his transfer targets.
Bilic is credited with getting Tore back on track after a shaky few years after making it out of the Chelsea academy.
Tore established himself as one of the most highly sought after players in the league under the stewardship of Bilic and the two built a close relationship.
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 former Besiktas manager also said that West Ham were not in direct talks which raised the question of whether negotiations are being being carried out through intermediaries.
“We are not in direct talks with Tore,” he added.
Tore has six goals and nine assists in all competitions for the Istanbul giants this season and played an important role in Besiktas winning the league title.
The 23-year-old winger also earned himself a call-up to the Turkish national side ahead of the European Championships in France this summer.