The 23-year-old was forced off the field in the first half of Wednesday's 2-2 draw with Liverpool after suffering a thigh strain.
And Wenger has confirmed that the injury will necessitate a long-term lay-off, telling reporters: "It will be four to six weeks [out].
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The 63-year-old has also defended back-up left-back Andre Santos, who has come in for criticism during a recent poor run of form.
Wenger added: "Santos is the type of player who gets sharper from competition. Yes [some criticism he gets is unfair].
"I believe we need two left backs. We had two before, we have two again now because Gibbs is out."
The Frenchman went on to explain that Nacho Monreal, who completed a move from Malaga on transfer deadline day, could be set for a debut at the weekend.
"Monreal is available for the weekend game. Yes [he can improve our defence] but we want him to take part in our collective play," he said.
"He can play as a left winger too and he can adapt very quickly."