Fabianski joined the Gunners five years ago but injuries and some high-profile errors gave Szczesny the opportunity to take over.
And he took the chance with both hands, becoming the undisputed first choice.
Fabianski told Polish paper Gazeta Wyborcza: "I hope that my adventure with Arsenal has come to an end.
"I intend to leave and play in a strong league. I wanted to leave in the winter, too, but Arsene Wenger did not give a green light."
Fabianski, who has 20 caps, was named the Polish league's top keeper in 2006 and 2007 during his time with Legia Warsaw.
Poland coach Franciszek Smuda has called him up to the provisional 26-man squad for the Euro 2012 championship, co-hosted by Poland and Ukraine, but he is expected to play second fiddle to Szczesny there too.
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