"Steven is, as reported, probably going to be (out for) up to about four weeks or so," Rodgers told the Merseyside club's website (www.liverpoolfc.com).
"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 had better news on Gerrard's fellow England midfielder Jordan Henderson, who suffered an ankle injury from a tackle by West Ham skipper Kevin Nolan. Nolan was sent off for the challenge.
"Jordan trained so he's fine," said Rodgers. "It was just precautionary the last few days. It was a bad challenge that was made on him, but thankfully he's okay."
Second-placed Liverpool, who are five points behind Premier League leaders Arsenal, visit Tottenham Hotspur on Sunday. Spurs are sixth but can join Liverpool on 30 points with a win.
(Writing by Ken Ferris; editing by Peter Rutherford)
