The England skipper limped out of Saturday's 4-1 win over West Ham with the problem and Rodgers on Thursday confirmed that the 33-year-old will not play again until the new year.
"Steven is, as reported, probably going to be up to about four weeks or so," Rodgers told Liverpool's official website.
"So he'll go through an extensive treatment programme over the next numbers 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."
If Gerrard misses the minumum four weeks then Liverpool will be without his services for clashes against Tottenham, Cardiff, Manchester City, Chelsea, Hull and Stoke, as well as their FA Cup clash against either Oldham or Mansfield.
Rodgers did add that Jordan Henderson, who hurt his ankle in a Kevin Nolan challenge last weekend, had been training without problem.