The Wales international hobbled out of Tuesday’s FA Cup fifth-round replay win at Hull, having lasted just 16 minutes following his introduction from the bench in place of Gabriel.
While Gabriel, and fellow defender Per Mertesacker, are likely to be available for the sixth-round tie at home to Watford on Sunday having both been forced off in the 4-0 victory at the KC Stadium, Ramsey faces a longer spell on the sidelines.
Asked if Ramsey could be out for the season, Wenger replied: “I don’t believe that.
“At the moment his clinical signs are quite positive. I would say four weeks. It was a bit worse than I thought because I didn’t see it.
“Usually when a player gets injured with a muscle, he stops and I did not have that feeling of precaution so I thought he would be available for Sunday.”
Ramsey’s injury rules him out of the upcoming Wales friendlies and means he also misses a key stage of Arsenal’s season – with the next week set to have a huge bearing on whether the campaign is deemed a success or failure.
The Gunners host Watford as they continue their quest to win a third successive FA Cup crown before travelling to Barcelona with the unenviable task of overturning a 2-0 deficit against the reigning European champions.
Wenger’s men then fly back to London before making the journey north to face Everton at Goodison Park in Saturday’s early Barclays Premier League kick-off – knowing they could be as many as 11 points adrift of leaders Leicester.
But Wenger is keen to keep all eyes on the next game, rather than wondering what might be later in the week.
“We want to stay alive in all of them (competitions) but then again lets be focused on the next one.
“I don’t believe anyone in my side is thinking about Barcelona at the moment, or Everton, they are completely focused on Watford.”
While Ramsey must sit out the most important of weeks, Wenger is hopeful of having Laurent Koscielny back from his calf problem – while Francis Coquelin returns to the squad having served his one-match ban for his sending off in the 2-2 draw at Tottenham last weekend.
“He has to join in normal training tomorrow and then for me to take a gamble on a guy that has run training sessions with the group or not,” Wenger said of Koscielny.
“After advice from medical people, I will take the gamble if he is available. Will I risk him or not? There is a little medical risk but I have not made a decision yet. Coquelin is coming back from suspension and (Hector) Bellerin was not in the squad (at Hull). He is now available.”