Liverpool said the injury "could keep him out of action for up to six weeks" starting with Saturday's league game against arch-rivals Manchester United at Anfield.
The news is a big blow for the Reds, whose form has suffered in recent weeks, with Klopp's men having won only one of their last seven matches in all competitions.
He has already missed three domestic matches this season after receiving a suspension following his sending off at Manchester City last month.
It was during that run of fixtures that concerns over Liverpool's accuracy in front of goal came into sharp focus - Jurgen Klopp's side had 79 shots in the games he missed, just 18 of which were on target.
Mané has scored three goals in four Premier League appearances.
They average 2.2 Premier League goals with Mane in the team, compared to 1.6 goals in his absence.
Liverpool are in seventh place now after an indifferent start to the campaign, seven points adrift of leaders Manchester City but only two behind Tottenham in third.
Senegal, who are top of their World Cup qualifying group, face two crucial games against South Africa on 10 and 14 November, which Mane is also likely to miss.