The prognosis means the 21-year-old is ruled out for the rest of the season and will also miss the European championship finals in his home country, which open with France playing Romania in Paris on June 10.
The injury came just as Chelsea, whose hopes of retaining their domestic title are effectively over after a poor campaign, prepare to play Paris St Germain next week in the last 16 of the Champions League.
Chelsea said the centre back, who has made a big impression this season, injured his anterior cruciate ligament when he fell awkwardly in the 1-1 draw against Manchester United at Stamford Bridge on Sunday.
"Kurt will undergo surgery in the next 48 hours and is expected to be out for approximately six months," the west Londoners said on their website (www.chelseafc.com).
Zouma has been on the fringes of his national team ahead of Euro 2016, earning his first cap last March and playing twice. He has made 32 appearances for Chelsea.
Since Guus Hiddink replaced sacked manager Jose Mourinho in December, he has started every game.
"I will have surgery this week & will come back stronger. Thank you everyone for all the messages," the player said on Twitter.
His absence could mean more regular starts for reportedly unsettled defender Gary Cahill, while the club might also want to re-consider the future of captain John Terry, who said recently he was not getting a new contract.
England international Cahill had fallen behind Zouma in the pecking order and was growing concerned about the lack of playing time impacting on his Euro 2016 ambitions.
Chelsea are at home to relegation-threatened Newcastle United in the Premier League on Saturday.
(Reporting by Alan Baldwin, editing by Pritha Sarkar and Mark Trevelyan)
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