Bernard, 28, has struggled to nail down a spot in the Everton side under Carlo Ancelotti and has been linked with a potential exit from Goodison Park this month.
Italian Serie A side Roma have been floated as a potential destination for Bernard, but now Turkish giants Fenerbahce are making a play for the wide-man.
Despite securing Mesut Ozil from Arsenal, Fenerbahce still want to make signings and, according to Turkish daily Takvim, have contacted Everton about Bernard.
Fenerbahce are keen to strengthen their options on the flank and want to take Bernard to Turkey.
They are also trying to bring forward a deal for QPR attacker Bright Osayi-Samuel.
The QPR man has signed a pre-contractual agreement with Fenerbahce and is due to join the Turkish giants in the summer when his deal runs out.
Fenerbahce will look to sign Osayi-Samuel this month and if they succeed will aim to have Bernard in the ranks for the start of next season.