The Brazilian midfielder is set to call time on his Liverpool career given his slump down the pecking order at the club, with Besitkas manager Senol Gunes having enquired about the 28-year-old's availability.
The deal, however, is with the view to a loan deal rather than a permanent transfer.
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Leiva has found first-team action hard to come by in recent seasons, making only 16 starts last season, and didn’t feature in either of Liverpool’s opening two fixtures against Stoke City and Bounremouth.
And with manager Brendan Rodgers preferring the likes of Jordan Henderson, James Milner and Emre Can ahead of him, a transfer away from Anfield looks imminent.
However, Besiktas won’t be his only option as the combative midfielder is attracting major interest from Serie A with both Milan based sides - Inter and AC Milan - also enquiring about his availability.
Leiva still has two years left on his current deal with the Reds and might bring the curtains down on his eight-year stay with the Premier League side having made 275 appearances in all competitions since 2007.