Fabianski was usurped as the Gunners' No. 1 last season by compatriot Wojciech Szczesny and has been restricted to just one start so far this season, in the Carling Cup victory over Shrewsbury last month.
With Wenger settled on Szczesny, Fabianski has cut an increasingly frustrated figure on the Arsenal substitutes' bench and believes he may have to move in order to secure a place in Poland's squad for next summer's European Championship finals in his homeland.
"If my situation does not change - I'll consider asking for a move," Fabianski said. "If I want to go to Euro 2012 the only way I can persuade the coach to take me is by playing on a regular basis."
The 26-year-old earned the moniker 'Flappyhandski' after some high-profile errors during his early days at Arsenal, which included a nightmare performance against Porto in the 2009-10 Champions League and errors in the Premier League against Stoke and Blackburn among others.
The position of goalkeeper has been problematic for Wenger in recent years and the Frenchman has never managed to recreate the stability that he enjoyed between the posts with David Seaman.
Jens Lehmann is the closest Wenger has come to finding a permanent successor to Seaman, but the 2006 UEFA Club Goalkeeper of the Year became erratic towards the end of his time at Emirates Stadium.
As well as Szczesny and Fabianski, the Gunners also have Italian Vito Mannone and Manuel Almunia competing for a place in goal, with the latter currently out on loan with West Ham.