The 28-year-old has failed to establish himself as Arsenal's number one during his near seven-year tenure at the Emirates Stadium, with the likes of Manuel Almunia and fellow Pole Wojciech Szczesny being Arsene Wenger's preferred options in that time.
Fabianski has made just four appearances for Wenger's side this season, with all of those coming in domestic cup competitions.
With his contract due to expire at the end of the season, Fabianski has decided his best option is to ply his trade elsewhere in order to secure regular first-team football and challenge Szczesny to be his country's first-choice keeper.
"I want to be number one in the club and in the Polish national team," he is quoted as saying by The Daily Star.
"I have learned a lot at London during very positive and challenging periods.
"With biggest respect to this great club, I didn't accept the extension offer as I am looking forward to a new challenge in England or abroad."
The news represents a further blow to Wenger, who saw his side fall to a crushing 5-1 defeat at Liverpool on Saturday.
Arsenal will look for a swift response when they welcome reigning Premier League champions Manchester United to the Emirates on Wednesday.
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