The defender was told he would be out for three months but Carragher, 33 in January, is determined to return sooner, something which manager Roy Hodgson hinted at earlier this week.
"Jamie is still nine weeks away from coming back, if we are to believe the doctors, but knowing Jamie it would not surprise me if those nine weeks became less than that," said the Reds boss.
Head of sports science and medicine Dr Peter Brukner admits the veteran centre-back is working hard to come back earlier.
"Jamie is making good progress and as you can imagine he is pushing things to the limit to get back but we are still on target for a February return," said the Australian.
"It still takes time for the surgery to heal and for him to get strong enough around the shoulder, but he's already working on his fitness and when he does come back he should be able to slot straight back in.
"Jamie is desperate to play and if it was up to him he would throw the sling away and get out there, but we are pushing as hard as we can to help Jamie but it does take time for these tissues to heal."