Gerrard suffered the problem in Saturday's 4-1 Premier League win over West Ham and Rodgers is resigned to losing the midfielder until the new year.
If the England skipper is ruled out for the minimum of four weeks, then he will miss clashes against Tottenham, Cardiff, Manchester City, Chelsea, Hull and Stoke, as well as the FA Cup third-round clash against either Oldham or Mansfield.
"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."
Rodgers should have fellow midfielder Jordan Henderson at his disposal after he returned to training following an ankle problem.