The Denmark international was substituted with the injury in the closing stages of an FA Cup third round win over Oldham Athletic last Sunday.
Agger posted on Twitter: "Disappointed with my injury! But I hope to be back strong within 4 weeks! Thanks for all the supporting tweets. ynwa."
The center back, who has been in and out of Brendan Rodgers' starting XI this season, is set to miss the Merseyside derby clash with Everton at Anfield on Jan. 28.
Agger also misses the club's next two games against Stoke City at the Britannia Stadium and at home to Aston Villa, as well as an FA Cup tie with either Bournemouth or Burton Albion.
But the Dane could be available in time to face Premier League leader Arsenal at Anfield on Feb. 8.
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