He had just begun to get back his best form, scoring in his previous two matches for club and country.
Wenger suggested Ramsey may have picked up the injury as a result of his busy schedule, having already made 17 appearances so far this season.
"He had a scan today but he's out. I believe he will be out until after the next international break. With the (last) international break, the fact that he played against Andorra certainly cost (Gareth) Bale and him as well," Wenger told the club's website (www.arsenal.com).
Former Tottenham Hotspur player Bale came under fire from Spanish media, with suggestions he had put Wales in front of his club side Real Madrid by playing against Andorra when Wales had already qualified for the Euro 2016 finals.
Ramsey will miss Saturday's game against Everton, as well as the North London derby against Tottenham Hotspur on Nov. 8.
Wenger also confirmed back-up goalkeeper David Ospina is likely to be out for the same duration.
(Reporting by Shravanth Vijayakumar in Bengaluru; editing by Martyn Herman)
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