The 32-year-old Brazil international was hurt during last weekend’s 1-0 victory over Crystal Palace and went off late in the second half.
Reports in the British press say tests have shown Alisson is not likely to be fit to play until after next month’s international break.
That would mean the first match he would potentially be available for would be the trip to Southampton on November 24.
Arne Slot’s Premier League leaders face Chelsea on October 20 and Arsenal a week later, while their next two Champions League fixtures pit them against RB Leipzig and Bayer Leverkusen.
Caoimhin Kelleher, who missed the Palace match through illness, will be expected to deputise, as he did last season when Alisson was ruled out with another hamstring injury.
Speaking after the victory at Crystal Palace, Slot predicted a lengthy lay-off for his number one goalkeeper.
“We have to wait and see but it will be a few weeks for him to be back. I think it is, yes (his hamstring),” he said.
“He’s clearly our number one, he’s the best goalkeeper in the world. It’s always a blow when he gets injured, not only for him but us as a team.”
Liverpool have 18 points from seven Premier League matches — one more than defending champions Manchester City and Arsenal.
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