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.
With midfield engine Michael Essien also missing after tearing knee ligaments four months ago, Scolari may be forced to bolster his squad before the January transfer window closes even though chief executive Peter Kenyon in November virtually ruled out new signings.
The Londoners face Serie A Juventus in the first knockout round of the Champions League.
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.
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