The Denmark international sustained the injury in the closing stages of Sunday's 2-0 FA Cup third-round win over Oldham at Anfield.
"Disappointed with my injury! But I hope to be back strong within 4 weeks! Thanks for all the supporting tweets," the 29-year-old wrote on Twitter.
Agger's absence is a blow to Reds boss Brendan Rodgers, whose injury list was showing welcome signs of easing.
However, Agger's unavailability will be offset by the progress made by Mamadou Sakho's recovery from a hamstring strain ahead of Sunday's Premier League trip to Stoke, while striker Daniel Sturridge is back in full training having been out since the end of November with a sprained ankle.
Captain Steven Gerrard played 77 minutes against Oldham to add to the half an hour he had coming off the bench against Hull City four days previously to show he had fully recovered from a hamstring strain.
"Over Christmas we were unfortunate we had a number of injuries which brought the squad down to about 15 players," said Rodgers.
"Thankfully over this last week we are starting to get one or two back and they are very important players for us.
"Steven came back into the fold against Oldham which was great and Daniel is back training now, which is brilliant for us."
Watch Liverpool take on Stoke at the Britannia live on Sky Sports 1HD at 4.10pm on Sunday.