The Turkish transfer window remains open until Friday, leaving the Black Eagles the chance to sign the Frenchman.
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The Daily Mail says Besiktas are turning to Meslier after an approach for Lucas Perri last week was rejected by Leeds.
Meslier, for his part, is open to the move after dropping to third choice this season behind Perri and Karl Darlow.
The 25 year-old is off contract in June, so Besiktas will be seeking to snap him up for a nominal fee.