The 23-year-old picked up a thigh strain during the first half of Wednesday's 2-2 fightback draw against Liverpool and was intially ruled out for three weeks.
But he now faces twice as long on the sidelines and is set to miss both Champions League last-16 ties against Bayern Munich.
"It will be four to six weeks (out)," Wenger told reporters on Friday. "I believe Gibbs will be out for the Champions League."
Andre Santos, whose recent performances have come in for criticism from supporters, will have to fill the left-back role against the Germans because new signing Nacho Monreal has already featured in the competition for Malaga.
"I (cannot) play Monreal," Wenger added. "Santos could have a key role."
Monreal, an £8.5million capture, will at least bolster Wenger's squad for Saturday's visit of Stoke.