Mane was taken off with a minute remaining in Senegal's 2-0 World Cup qualifying win over Cape Verde Islands on Saturday.
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The influential forward - and Ballon d'Or nominee - could miss up to six weeks after suffering the injury on global duty with Senegal, the club have revealed.
Liverpool won just once in seven matches in all competitions while Mane was at the African Nations Cup in January.
Mane's absence is a huge blow for Liverpool boss Jurgen Klopp after the forward scored three goals in four Premier League appearances this season.
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Mane had already endured a turbulent start the season after serving a three-match domestic suspension having received a red card during his side's 5-0 trouncing at City in September.
Mane was included in the 30-man shortlist for the 2017 Ballon D'or which was released on Monday.
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