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.
- Sports & Recreation
- the Premier League