The Black Eagles are reported to be preparing a January bid for the Colombian international. Beşiktaş were closely linked with Ospina over the summer but he decided to stay and fight for his spot in the Arsenal team.
Ospina fell down the pecking order at Arsenal following the signing of Petr Cech but he was expected to get some playing time after Wojciech Szczęsny was loaned out to Roma.
The Colombian keeper reportedly had a verbal agreement with Wenger that he would get game time in the Champions League despite the arrival of Cech from reigning Premier League champions over the summer.
Arsenal boss Arsene Wenger did play Ospina in the early stages of the Champions League but he was dropped after mounting pressure to play Cech after making a series of costly mistakes.
Beşiktaş are currently first in the Turkish Super League and on course to booking a place in the Europa League Round of 32.