Aaron Ramsey and Robin van Persie back in training next week, however
The young full-back was substituted after 18 minutes of the 4-0 League Cup win at Newcastle United on Wednesday after a collision with Wayne Routledge.
Gibbs suffered a metatarsal injury in November last year that ruled him out for the rest of the season, and Wenger believes he is likely to spend another significant spell on the sidelines.
He told the club's official website: "We lost Kieran Gibbs last night.
"That is the bad news. It is a medial knee ligament strain.
"It needs to be checked on the scan tomorrow but it will be a few weeks."
Wenger will also be without Manuel Almunia, Thomas Vermaelen, Aaron Ramsey and Robin van Persie for the game against West Ham United this weekend.
"We have nobody else back from Manchester City," he added.
"Van Persie and Vermaelen are still out, Ramsey and Almunia are not ready, so it should be a similar squad to the one that played at Man City."
But there is good news in the shape of Aaron Ramsey and Robin van Persie.
Young Welsh star Ramsey suffered a long-term injury at Stoke City last season, but Goal.com UK exclusively revealed that it is hoped he will be back in action in 10 days.
Wenger added: "Van Persie and Ramsey are very close to coming back to full training now.
"They should be back in the group at the end of next week.
"Of course we should not set the same [recovery] time to Ramsey as Van Persie.
"Aaron will need a longer period to be completely match-fit for the first team, but Van Persie is not far away now.
"Manuel Almunia is maybe one to two weeks away. With Vermaelen, it is difficult to set a fixed time because the Achilles improves slowly."