The Wales international, 23, has been one of the stand-out performers in the Barclays Premier League this season and played a key part in helping the Gunners into a position to challenge for their first league crown since 2004.
Ramsey suffered the injury in training and Arsenal manager Arsene Wenger admitted he may now need to bring in a midfielder as cover in the last hours of the January window, with the club having been heavily linked with Schalke's Julian Draxler throughout the month.
In quotes carried by the club's official Twitter feed, Wenger said: "Aaron Ramsey will be out for four to six weeks. I count six weeks.
"We are still trying to bring one body in because of Ramsey's setback and Flamini's suspension."