The 29-year-old has missed the start to the season after picking up the problem during the Reds' pre-season friendly clash with Atletico Madrid in August and was expected to be out for three months.
However, the England international has made good progress with his recovery and is expected to start running outside next week with the view to re-joining his team mates in full training during October's international break.
Boss Jurgen Klopp will be happy to have Lallana back, the former Southampton youngster having been key to his plans last season.
Klopp said: “Adam made the Champions League squad because it was a moment when we had to think about these things and the medical department told me that in around about six weeks he should be back.
“That was around August 31. So it should be about another four weeks before he’s normal training. It looks all on the right way for him.”