The 23-year-old England international picked up the injury during Wednesday's 2-2 draw with Liverpool in the Premier League.
"It will be four to six weeks [out]," Wenger told a news conference.
Wenger has extra cover for the position having signed Spain left back Nacho Monreal from Malaga on Thursday.
Arsenal, who host Stoke City on Saturday, are sixth in the Premier League, four points behind the Champions League places.
Brazilian Andre Santos, who has struggled to impose himself in the Premier League, is another option at left-back and may have to play in the Champions League last-16 tie with Bayern Munich on February 19 and March 13.
Monreal has already played in this season's Champions League for Malaga and will therefore be ineligible but he could make his debut this weekend.
"He is available for the Stoke game. I have not made a decision yet," Wenger said.
"We bought Monreal as a left-back but he can play as well as a left-sided midfielder. We bought him as well because we believe he can integrate into our style of play and adapt very quickly to that.
"He has the characteristics to play our game because he is very mobile, has a good left foot and is very good in combination play. That is vital in our style of play."
Arsenal head to Stoke boosted by the return of midfielder Mikel Arteta following a calf strain. However, defender Thomas Vermaelen is doubtful with an ankle injury and Francis Coquelin is close to a return but this weekend could be too soon.
One noticeable absence from Arsenal training was David Beckham, who had this week been working on his fitness at the club while he was a free agent after leaving Los Angeles Galaxy.
Wenger jokingly accepted credit for the former England captain's move to Paris Saint-Germain announced on Thursday.
"It just goes to show you how efficient we are with our training sessions," he grinned. "He came here to practise once and straight away was bought by Paris Saint-Germain."
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