The 23-year-old Wales international has been one of the London club’s star performers this season, but he has not played since December 26 due to the problem.
He returned to full training earlier this week, only to aggravate the injury, which prevented him from making a comeback in Arsenal’s 2-2 draw at Southampton on Tuesday.
As a result, and with Arsenal having conceded top spot in the Premier League table to Manchester City, Wenger has revealed that he will try to sign a new midfielder before the January transfer window closes later on Friday.
“It is certainly a few weeks. It will be between four to six weeks. I count six, so I have no bad surprise,” Wenger told a press conference.
“Of course it’s a blow for us because we recently lost (Theo) Walcott for a long period, now Ramsey is out for a long period and it is a big setback.”
Wenger said he was “80 percent” confident of bringing in a new player before the transfer deadline falls at 2300 GMT, but he ruled out a move for Schalke’s Julian Draxler, with whom Arsenal have been linked in the media.
“The Draxler situation has been created by the newspapers, not by me,” said Wenger.
“You ask me, what is that situation? I say nothing is happening on that front. He will stay at Schalke.”
Wenger added: “We are open-minded and active. We look at the opportunities.
“We are still trying to bring at least one body in because we are a bit hit now by the red card for (Mathieu) Flamini, the setback of Ramsey and the fact that (Jack) Wilshere is not available, which I hope will be short term.
“We are in an area where we had plenty of players, suddenly we are a bit short. It is 80/20 and I am confident we will be able to do something today (Friday).”