The 29-year-old fullback is likely to miss the rest of the Premier League season, which ends on May 9, and faces a race to be fit for the World Cup in South Africa starting on June 11.
Cole, who has been England manager Fabio Capello's first choice left back, will be sidelined for their warm-up match against Egypt on March 3 at Wembley Stadium.
Asked if he would be fit for the World Cup, Chelsea manager Carlo Ancelotti told Sky Sports News: "I think so. I hope so for him and the for the England team."
Chelsea will be without Cole for their Champions League last 16 first-leg tie at Inter Milan on February 24 and the second leg on March. 16 when the club's former manager Jose Mourinho returns.
Cole suffered the injury just before the hour mark in the 2-1 loss at Goodison Park in a challenge with Everton's United States forward Landon Donovan, a player he could face in South Africa in the Group C opener on June 12 in Rustenburg.
Ancelotti said Chelsea would do all they could to help Cole get back to full fitness as soon as possible, though the Italian added that the club had cover for his position in the squad.
"Now we have to do our best to help him recover quickly (but) Yuri Zhirkov is a fantastic fullback so we can use him," he said referring to the Russia midfielder who usually plays in an attacking role but can also fill in at left back.
Cole, who has 77 caps, is the joint-second most capped player in the current England squad along with vice-captain Steven Gerrard behind David Beckham (115). His understudy is former Chelsea team-mate Wayne Bridge of Manchester City.
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