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Arsenal are demanding a £4.5m [€5m] transfer fee from Besiktas for David Ospina according to the Sabah newspaper.
Per the source, the Black Eagles want the Gunners to agree to a 1.8m [€2m] deal.
Talks are set to continue this week after personal terms between Besiktas and Ospina were agreed.
The Colombian international has been given the green light to leave by head coach Unai Emery.
“[On Thursday] the window closed here in the Premier League,” Emery said.
“Today, we had a meeting with the players to close all the possibilities to bring more players. We only have three players: Carl Jenkinson, David Ospina and Joel Campbell, with the possibility to leave.”
Turkish-Football reported that Besiktas agreed personal terms with Ospina last week.
Besiktas president Fikret Orman travelled to London on Tuesday with club officials where talks were held with Arsenal over the 29-year-old.
Ospina was left out of the matchday squad by Emery against Manchester City on Sunday and is not expected to feature this season if he stays on at the North London outfit.
The Colombia international has another two-years left on his contract but has dropped down the pecking order at Arsenal behind Bernd Leno and Petr Cech.
The Black Eagles would meanwhile, be able to offer first-team football this season after selling Fabri to Fulham last month.
Current first-choice keeper Tolga Zengin would drop back into second-place keeper again should the World Cup star join.