The France international striker picked up a seemingly innocuous injury against Everton at the weekend, but was subsequently sent to a specialist amid fears that he had broken his ankle.
And Wenger has now revealed that he does indeed face a lengthy spell on the sidelines, though he maintains that he is happy with his current striking options.
He told Sky Sports: "The news is that he had surgery today, that surgery went well, but he will be out for - maybe to be competitive at the top, top level - end of December, beginning of January."
Asked whether he will sign a striker to cover for Giroud's injury, Wenger added: "Look, I don't know yet. We try to look around but we want quality and we have quality.
"We have fortunately Walcott coming back soon. I believe Sanogo will have an impact this season. We have Podolski, we have Campbell who can play center forward.
"To find players who are better than the players we have would be very difficult now."
Wenger did suggest that he will weigh up his options but insists that 33 million pound club-record signing Alexis Sanchez was bought to play as a striker, while also praising the likes of Yaya Sanogo and Joel Campbell.
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"We will consider [it]," he said. "Just in England you cannot, every time you have a problem, always buy. If you have an injury after the first of September, you have to live with it.
"I bought Sanchez to be a striker, and I think Walcott can be a very good striker. I think Sanogo is a very good striker. Podolski can play there. Campbell, if you ask him what he is, he's a striker.
"They are all players of quality and we'll see."