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 centre-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 club-record signing Alexis Sanchez was bought to play as a striker, whilst also praising the likes of Yaya Sanogo and Joel Campbell.
"We will consider [it]," he said. "Just in England you cannot, every time you have a problem, always buy. If you have an injury afte rthe 1st 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."