Unai Emery is attempting to trim his squad down before the deadline passes with the Gunners stacked in the goalkeeping department.
Summer signing Bernd Leno is set to compete with Petr Cech for the number one jersey, with Emiliano Martinez set to be the third-choice reserve.
Colombia international Ospina is prepared to depart the Emirates Stadium and has even settled on personal terms with Besiktas, according to reports from Turkey.
However, Arsenal are insistent on getting the the 29-year-old off their books on a permanent deal but Besiktas are pushing for an initial loan deal.
Reports from Italy earlier in the week also claimed Arsenal had rejected a similar loan deal for Ospina from Napoli.
Gunners striker Danny Welbeck is also on the clubs transfer list, with Besiktas credited with interest in him, but the report claims the England strikers future was not discussed in Arsenals meeting with Besiktas on Thursday.