The Slovakia captain, who represented his country at this summer's European Championships, joined Liverpool from Zenit St Petersburg in 2008.
Skrtel, 31, made 21 starts in the Premier League last season, but Jurgen Klopp has added to his defence this summer with the signing of Joel Matip.
Fenerbahce finished second in the Turkish top-flight last season.
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Fenerbahce have confirmed the club are in talks with Liverpool over the transfer of defender Martin Skrtel.