The centre back, who has made a positive start at Anfield after joining on a free transfer in the close season, was expected to be sidelined for several weeks, but Rodgers said he will be assessed ahead of the weekend clash.
"His injury is nowhere near as bad as what we thought," Rodgers told reporters.
"We'll just assess that over the next 24/48 hours.
"I think we were all thinking that he might be out for longer after the game, but he's made an excellent recovery. The diagnosis isn't anywhere near as bad as we thought, so we'll see how he is for this weekend."
Aly Cissokho, a close-season loan signing from Valencia, was substituted after just nine minutes of the cup match and Rodgers said he would now be set for an extended spell on the sidelines.
"He could be out for four to six weeks, which is a blow because he came in and trained really well last week," Rodgers added.
"He's on crutches at the moment and we'll just look at that over the coming weeks. But he certainly won't be available (for the Manchester United game)."
Champions United will provide Liverpool, who have won both their opening Premier League matches, with the first serious test of their top-four credentials.
Liverpool lost home and away to their fierce north-west rivals last season on their way to a disappointing seventh place league finish.
Rodgers was keen to point out, however, that the club had enjoyed a recent upturn in fortunes.
"If you look at our numbers since we last played Manchester United in January, that's the improvement where we've really kicked on. The big thing for us was consistency," he said.
"From when we played them in January, we've played 18 Premier League games each - they have won 11, we have won 10; we have drawn six games and they have drawn five; we have both lost two games.
"So over the course of that period from January they have two points more than us. Obviously they became champions. For us, it's a case of continuing with that consistency. If we can do that, I think we can compete with anyone."
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