The goalkeeper will miss Saturday's North London derby at Tottenham Hotspur as well as the FA Cup replay against Hull City, Premier League games against West Bromwich Albion and Everton, and the second-leg tie in the last 16 of the Champions League against Barcelona.
He will be replaced by reserve keeper David Ospina, Wenger told a news conference on Friday.
Centre back Laurent Koscielny will also miss the game against Spurs with a minor calf injury and is doubtful for Tuesday's trip to Hull but is likely to return to face West Brom on March 12.
Wenger rejected former Arsenal striker Thierry Henry's comments that fans were enraged by the loss to Swansea.
Arsenal's record goalscorer said he had never heard the Arsenal supporters as angry as they were at the Emirates on Wednesday.
"Thierry Henry has his opinions," Wenger said.
"He has not found the measurement of the fans' angriness, of 60,000 people, straight away because he sits in the best seats in the stadium."
(Reporting by Simon Jennings in Bengaluru; Editing by Ed Osmond)
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