Southampton manager Ronald Koeman confirmed the injury following his side's 1-1 draw with Ipswich Town in the FA Cup third round on Sunday.
Mane missed the game having limped out of the 2-0 victory over Arsenal in the Premier League on New Year's Day and Koeman revealed he will not be available for the continental showpiece, which begins on January 17 in Equatorial Guinea.
"Sadio has a calf muscle injury," he told the club's official website. "He will be out for four-six weeks and he don't (sic) play the Africa Cup.
"Maybe the [Senegal] Federation will call him to come and see the player, but he will not play and will be out for at least four weeks."
Mane has shone for Southampton this season since his move from Red Bull Salzburg, scoring four goals in 14 Premier League matches.
His absence was felt as Southampton struggled to convert their chances against Ipswich, with the Championship side holding on for a replay.
Southampton were behind after 19 minutes at St Mary's as Darren Ambrose gave the visitors the lead.
Morgan Schneiderlin levelled things up before half-time but Koeman was left frustrated.
"I am disappointed about the final result," he said. "It means a replay and already we play a lot, it wasn't what I wanted today.
"We didn't start the game well, we didn't have enough concentration, and they were totally focussed so the difference was not so big, and it's always difficult to change that.
"I am happy about how we played the second half, we had total control and a lot of ball possession.
"Maybe we didn't have the quality to break it open and create possibilities, but that is part of football."