The Daily Mail says Szczesny is facing a fight to save his Gunners career after a string of below-par displays. The Pole was dropped in favour of countryman Lukasz Fabianski for Arsenal's last two games against Bayern Munich and Swansea.
The Gunners kept two clean sheets, which is likely to result in an extended spell on the bench for Szczesny.
In an interview with Gazeta Wyborcza ahead of Arsenal's win at the Liberty Stadium, Szczesny said: "I am too happy [here] to go somewhere else. I believe in the philosophy of the club and that Arsenal will start to win again. I have a debt to pay here for the club. I do not want to go anywhere else. As long as they want me here, there is no "somewhere else". I have a contract until June 2015."