Kelly was forced off the final minutes of the 2-1 defeat against Sir Alex Ferguson's team and has revealed that he will not return to action until "next year".
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He tweeted: "What I can say is its not the worst injury I've had, and for sure ill be fighting for my place next year when I'm back fitter and stronger!"
Earlier on Monday, Kelly was pessimistic about his chances of a quick comeback, posting: "Hoping for the best but fearing the worst :( got my scan at 8 then will let you guys know."
Kelly's team-mate and fellow defender Daniel Agger could also be set for a spell on the sidelines after suffering a suspected medial ligament injury to his knee in the second half of the match at Anfield.
Rodgers confirmed after the match that Fabio Borini, meanwhile, had a swollen ankle. Liverpool are still searching for the first win of the campaign and sit in 18th place in the Premier League.