Ospina has struggled for game time since arriving from Nice after the 2014 World Cup.
The Colombia international has spent most of his time in north London behind Petr Cech in the pecking order.
And The Sun claim the 28-year-old is determined to force a move away from the Emirates in the summer.
However, his departure could see Szczesny return to Arsenal.
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The Poland international was the club's first-choice goalkeeper before Arsene Wenger signed Cech from Chelsea in 2015.
He has spent the last two seasons on loan at Serie A giants Roma, helping them up to second in the league table.
Should Szczesny make clear his intentions to leave the Gunners permanently, then Wenger might promote third-choice Emiliano Martinez to be his back-up.
All three of Arsenal's senior goalkeepers face uncertain futures.
While Cech is the club's No1 for now, the 34-year-old has shown signs of waning this season.
As a result, the Gunners have been backed to make a move for Sunderland star Jordan Pickford.