The Turkish newspaper Fanatik reports that the Gunners are willing to offload the goalkeeper over the summer and that Besiktas are close to agreeing upon on a €3.5 million deal.
The report goes onto state that the Istanbul giants also made an offer for Ospina bacl in January but both times the deal fell through.
Arsenal may however, be keen on keeping Ospina on if on-loan keeper Wojciech Szczesny decided to stay at Roma on a permanent move.
Besiktas won the Turkish Super League title last season and booked a direct place in the Champions League group stage.
Ospina made 12 appearances for the Gunners this season and is expected to star for Colombia at this summers Copa America tournament.