The England international sustained the injury during the club's 1-0 victory over Manchester United at Anfield, during a challenge with Patrice Evra.
Johnson missed his country's FIFA World Cup qualifiers against Moldova and Ukraine as a result, but Rodgers is hopeful the 29-year-old will return to the first-team fold in October.
"It's a disappointment because I think he's made a good start and he was excellent against Manchester United," Rodgers said ahead of Liverpool's trip to Swansea City on Monday.
"Fortunately for us, with the way it works out and the international breaks, he'll probably only miss about four Premier League games.
"Glen is the type of player that you never realise how much you miss him until he's not there. He'll go away, work hard and keep himself in the best shape he can."
Johnson himself is confident of a quick recovery, adding on Twitter: "Injury doesn't seem as bad as first thought, hopefully be back sooner than originally expected."
Despite the setback to Johnson, Rodgers is confident of having key trio Daniel Sturridge, Joe Allen and Kolo Toure available for selection against his former club.
Sturridge was unavailable for England duty due to a thigh injury picked up before the United game, while Toure and Allen were both injured during the League Cup victory over Notts County in August.
"He (Sturridge) was obviously disappointed to miss out on England games," Rodgers continued. "I felt for him because he came here, first and foremost, to do as well as he possibly can for Liverpool - knowing if he could do well, his aim was obviously to be the England striker.
"He returned back and there was a slight improvement, but obviously Roy (Hodgson) and his staff have to make the decision on whether he's going to be right or available. Obviously at that stage, he wasn't.
On Allen and Toure, the Ulsterman added: "Joe has trained with the team. He looked good (Friday) morning and is certainly fighting for a place on Monday.
"Kolo is with our physiologists and is working well. He should be available for Monday."
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