The report goes on to claim that the Yellow Canaries have agreed to a contract worth €2.2 million per-season with the Colombian international.
Ospina is currently on international duty with Colombia at the Copa America. The 26-year-old keeper kept a clean sheet in the 1-0 victory over Brazil in Group A on 18 June.
The Colombian international could fall down the pecking order if Arsenal complete the signing of Petr Cech from Chelsea.
Ospina would have to compete with Wojciech Szczesny to become the Gunners second choice keeper.
The 26-year-old joined Arsenal on a €4 million move from Nice last season and kept 10 clean sheets in 23 appearances for the North London based club.
Fenerbahçe have been in the market for a new goalkeeper after recently appointed sporting director Giuliano Terraneo deemed Volkan Demirel not up to par.
The former Lazio and Inter Milan sporting director is reported to have close links with Arsenal boss Arsene Wenger.
The Yellow Canaries have been closely linked with Real Madrid keeper Iker Casillas but have been put off by his wage demands and recent form.
Fenerbahçe finished last seasons as runners up in the Süper Lig and booked a place in the Champions League play-off stages.