The report claims that Hammers boss Slaven Bilic has made signing Gokhan Tore and Ozyakup a priority.
Bilic coached both players during his two-year stint at Besiktas and has a close relationship with both players.
The East London based club are still locked in talks with Besiktas over Tore. West Ham have offered £12 million but Besiktas want £17 million and negotiations are ongoing.
Besiktas are willing to part ways with Tore having already secured replacements and the club have been planning for his exit since January.
The Black Eagles are however, keen on keeping Ozyakup at the club for another season and club president Fikret Orman has revealed that the star midfielder is not up for sale unless an extraordinary offer is made.
Ozyakup is no stranger to Premier League football having risen through the youth ranks at Arsenal before joining Besiktas in 2012.
The former Gunner has established himself as one of the most highly sought after players in the Turkish league playing a key role in Besiktas winning the league title this season.