The 32-year-old suffered the injury during the first half of Monday's FA Cup quarter-final replay win over Manchester United.
After the match, Blues manager Rafa Benitez confirmed the injury, adding: "This kind of injury, at least two weeks (out)."
And he joked: "That means for us maybe 20 games or something."
England international Cole departed after 21 minutes of the contest and will now miss Thursday's Europa League quarter-final first leg with Rubin Kazan and Sunday's Premier League match with Sunderland.
He is also a major doubt for the FA Cup semi-final with Manchester City at Wembley on 14 April.
The loss of Cole comes at a bad time for Chelsea, for whom the United clash was the second of six games in 16 days.
The Blues are already without England defender Gary Cahill who is out for "another 15 days at least" because of injury.