Chelsea interim boss Rafael Benitez confirmed the blow for Cole following the 1-0 victory, secured by Demba Ba's spectacular second-half strike.
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 April 14.
Asked about Cole, Benitez said: "This kind of injury at least two weeks.
"That means for us maybe 20 games or something."
Benitez must manage his squad through a congested period - the United clash was the second of six games in 16 days.
Cole's injury is the latest defensive blow, with England defender Gary Cahill out for "another 15 days at least", Benitez said.
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