The Turkish sports newspaper revealed that the Black Eagles are keen on the Colombian international and will submit a loan offer.
The report also claims that Besiktas are willing to match Arsenal’s £3 million asking price if they want to sell him on a permanent deal.
Besiktas are reported to have agreed personal terms with the 27-year-old old keeper and will be able to offer regular first team and Champions League football.
The Black Eagles almost signed Ospina last season but Arsene Wenger blocked the move and wanted the Colombian keeper to stay as the Gunners second choice keeper.
Besiktas president Fikret Orman has already confirmed interested in the Arsenal keeper and the Black Eagles have made improving defensively a priority over the summer after conceding the most goals in the top three – despite winning the league title.