Ospina is reported to be unhappy at the club with his lack of playing time and is considering his options.
The Colombian international is currently playing second fiddle to first-choice keeper Petr Cech in the Arsenal pecking order.
Besiktas have already held talks with Ospina’s agent and are keen on signing the 27-year-old ahead of their Champions League campaign.
Besiktas are preparing a £3 million bid for Ospina according to the Daily Star.
Besiktas also made an offer for Ospina last season and in January but both times the deal fell through.
The Black Eagles won the league title last season and booked a direct place in the Champions League group stages.
Ospina will be playing in the brand new 41,903 seater Vodafone Arena if he were to make the move and would be first-choice.