Lallana, who left Southampton earlier this month for £25m was hurt on the pre-season tour of the United States.
"No specific time frame has been placed on a return to fitness but injuries of this nature can result in a spell of anywhere up to six weeks on the sidelines," the club said.
The 26-year-old Lallana, a member of England's World Cup squad in Brazil last month, is also likely to miss away league games against Manchester City and Tottenham Hotspur.
Meanwhile Liverpool are on the brink of making Dejan Lovren their third signing of the summer from Southampton after Lallana and Rickie Lambert.
Lovren is due on Merseyside for his medical on Saturday ahead of a £20m move.
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