The 23-year old left-back limped off late in the first half of the Gunners' 2-2 draw with Liverpool at the Emirates Stadium and will be out for almost a month with a thigh strain.
"Gibbs has a thigh strain and will be out for three weeks. It's a big blow for us," Wenger said in his post-match press conference.
"He was an important player for us going forward and the game we wanted to play. It's a shame he's out for the next three weeks."