Cissokho sustained the problem early on in Tuesday's 4-2 extra-time Capital One Cup second-round victory over Notts County at Anfield, in which he was making his first start since joining on a season-long loan from Valencia.
Two other Liverpool players picked up injuries in the match, with fellow defender Kolo Toure being carried off on a stretcher due to a groin issue and midfielder Joe Allen limping off with a hamstring strain, and Rodgers had more positive news on that pair.
Asked about the trio at his press conference to preview Sunday's Barclays Premier League home clash with Manchester United, Rodgers said: "Toure's injury is nowhere near as bad as we thought.
"We will assess that over the next 24 to 48 hours. I think we were all thinking 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.
"If it's not this weekend, then we would think after the international break he won't be too far away.
"Joe likewise - it was only a grade one strain, so he may be doubtful for the weekend but certainly after the international break, he'll be fine.
"Aly Cissokho will be slightly different - he could be four to six weeks, which is a blow because he came in and trained really well last week.
"But he is on crutches at the moment and we'll look at that over the coming weeks. But he certainly won't be available."
Meanwhile, Rodgers was refusing to give anything away regarding various players Liverpool are reportedly trying to sign ahead of Monday's transfer deadline.
The manager was asked about five reported targets in Tiago Ilori, Andriy Yarmolenko, Shay Given, Mamadou Sakho and Victor Moses.
He said: "You guys keep asking and we as managers have to give the same answers to you.
"There is nothing to report until you have a player signed and in the building, and obviously because of the nature of the window and, certainly in this country, the nature of the gossip and speculation that goes on, there are so many players linked with the football club that if you answered every single question, you'd be here all day and all night.
"You are better to say nothing until the player is in the building.
"Until that point it is only ever speculation."