A statement on the Chelsea website said Cole had undergone successful reconstructive surgery on Saturday.
"I am very sad for Joe to have suffered this injury and I am disappointed to have a lost a very important player for me and the club at such a vital moment," manager Luiz Felipe Scolari told the website.
"I believe that with his spirit Joe will come back even better than before."
Cole's injury is a major blow to Scolari who has come under pressure in recent weeks after his side suffered a dip in form culminating with last weekend's 3-0 defeat at Manchester United.
Cole's season has been plagued by injuries with foot and calf problems restricting him to 20 appearances this season.
As well as being a key component in Chelsea's midfield, Cole is a regular in Fabio Capello's England squads, coming off the bench to score twice against Andorra in a World Cup qualifier in September. He will miss the home match against Ukraine in April.
