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The 31-year-old joined the Super Lig Club two years ago on a free transfer from Liverpool, but was frozen out last season by manager Giovanni van Bronckhorst.
Ole Gunnar Solskjaer's arrival in January did little to improve his fortunes, and his time at the club was plagued by injury.
He made just 18 league appearances last season as Besiktas finished fourth in the top flight, finishing a huge 33 points off champions Galatasaray.
The Southampton youth product also endured the humiliation of being left out of the club’s Europa League squad for the second stage of their campaign.
Overall, he featured in 50 matches for Besiktas, scoring just five times.
The England international made just 18 league appearances last season.
With the Reds, he won the Champions League and Premier League, but his final season was tainted by form and fitness issues.
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He is now expected to be linked with Premier League clubs.
However, the Serie A club is thought to be concerned about his wage demands.
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