Turkey star Gokhan Tore joined Besiktas from Rubin Kazan in 2014
The 24-year-old spent two years with Saints' Premier League rivals Chelsea as a youngster, before spells with Hamburg and Rubin Kazan led him to join Besiktas in 2014.
He has since excelled with the Turkish Super Lig giants, netting 19 goals and providing 26 assists in 111 appearances in all competitions.
And Hammers boss Slaven Bilic has been linked with a move for Tore, having managed him during his two-year spell with Besiktas.
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Tore is also thought to be keen on moving to a Champions League club and, despite their strong season, Southampton's sixth place finish means they can only offer second tier European competition in the form of the Europa League.