The Denmark international suffered the setback during Liverpool's 2-0 FA Cup victory over League One outfit Oldham last Sunday.
Agger was already ruled out of this weekend's Premier League visit to Stoke City as club officials awaited confirmation on the severity of the problem, but he is now set for a longer period on the sidelines and will miss the crucial Merseyside derby with Everton on January 28, while his involvement in the home fixture with leaders Arsenal on February 8 is in doubt.
Losing Agger comes as a blow to manager Brendan Rodgers, who is down to two fit centre-backs in the form of Martin Skrtel and Kolo Toure, as Mamadou Sakho continues his recovery from a hamstring injury sustained in the 2-1 defeat at Chelsea on December 29.
"Disappointed with my injury! But I hope to be back strong within 4 weeks! Thanks for all the supporting tweets," Agger posted on Twitter.
The 29-year-old has been linked with an Anfield exit during the January transfer window, with Serie A side Napoli - coached by former Liverpool manager Rafael Benitez - thought to be among his suitors, although Agger has refuted those claims.
Agger has made 13 appearances for Liverpool this season in all competitions.
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