The goalkeeper has grown frustrated at his lack of regular game time at the Emirates.
At the age of 28, Ospina is looking to secure first-team football away from the Gunners, with Petr Cech continuing to pull rank under Arsene Wenger.
According to Turkish outlet Fanatik, the Colombia international has already spoken with Fenerbahce ahead of the summer.
David Ospina has been limited largely due to just playing in cup competitions for Arsenal
The Turkish giants are expected to make an initial approach worth around £2.5m
Only a fee now needs to be agreed with Arsenal, who spent £3million on him to sign the shot-stopper in 2014 from Nice.
Fenerbahce are ready to sit down and negotiate with the north London outfit.
And it is said the Turkish giants are expected to make an initial approach worth around £2.5m.
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Arsenal may hold out for more, but with Ospina pushing to leave and terms sorted at Fenerbahce, a fee is likely to be agreed.
Wenger must find a long-term solution in goal with Cech ageing and Ospina going.
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A new keeper is said to be on Arsenal’s agenda this summer, with Sunderland’s Jordan Pickford and Stoke’s Jack Butland targets.