The Black Eagles will turn their attention to Ospina after the deal for Fabricio Agosto Ramirez fell through due to injury.
The negotiations are expected to go ahead some time over the next few days in London with Besiktas representatives expected to make a trip to England.
The Istanbul giants were closely linked with Ospina last season and over the winter transfer break but were unable to complete a move for the experienced keeper.
Ospina is reported to be unhappy with his role at Arsenal playing second-fiddle to first choice keeper Petr Cech and is expected to make a decision on his future following the end of Copa 2016.
The 27-year-old played a key role in Colombia reaching the semi-finals conceding just one goal in regulation time over the course of the tournament.
Besiktas president Fikret Orman recently that he wants Ospina at the club: “I do want Ospina and we are going to get a foreign keeper,” he said in a recent press conference.