The Colombian goalkeeper wants a move away from Arsenal in search of first team football, something he hasn't been getting at the Emirates Stadium ever since arriving from Nice back in 2014.
Ospina, 27, plays backup to Petr Cech for the north London giants, but before that, was seen as an understudy to Wojciech Szczesny, meaning his game time in England has been limited.
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The Mirror reports that Besiktas are interested in a £3 million purchase of the shot-stopper, which could end the Colombian's two-year association with the Premier League club.
Ospina joined the Gunners in the summer of 2014, following the departure of Lukasz Fabianski, who chose to join Swansea City on a free transfer.
Ospina has endured a lacklustre spell at Arsenal, where his chances have been limited. He joined the Gunners from French side Nice after an impressive display for Colombia at the World Cup in 2014.
He was touted as the long-term solution to Arsenal's goalkeeper situation, but just one year after signing him, Arsene Wenger turned to four-time Premier League Golden Glove winner and the club's sole summer signing of 2015, Petr Cech, who now plays ahead of the Colombian.
Ospina is at the phase of his career where he must look for regular first team football. For a man of his talent, he can't stay on the bench too much longer, and although Wenger would be more than happy to see him stay with the Gunners, it seems unlikely that will happen.
Besiktas were unsuccessful in securing the services of the Colombian on two occasions, the summer of 2012 and 2015, but despite suffering two setbacks in the past, they are still keen on signing him.
Wenger will have a decision to make this summer regarding his goalkeepers. Cech will almost certainly remain the Frenchman's number-one, but a decision will then need to be made regarding Ospina and Szczesny, with the latter spending the season on loan in Italy with AS Roma.
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