The 33-year-old had been an ever-present for the Reds this season in the Premier League but picked up the injury last time out in Liverpool's 4-1 win over West Ham.
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"Steven is, as reported, probably going to be up to about four weeks or so,” he told the club’s official website.
"So he'll go through an extensive treatment programme over the next number of weeks and hopefully we can get him back that little bit sooner.
"But certainly at this stage, it was a four to six-week injury and we'll see how we look with that."
Rodgers also had good news regarding fellow midfielder Jordan Henderson who picked up an ankle injury against the Hammers.
"Jordan trained so he's fine," he added. "It was just precautionary the last few days.
"It was a bad challenge that was made on him, but thankfully he's okay. But we'll see how he is tomorrow as well."