Carrick turned his ankle in United's League Cup penalty shootout loss to Sunderland on Wednesday after landing awkwardly on the astroturf strip that runs along their Old Trafford pitch.
The England midfielder was expected to be out for two weeks, but initial tests have shown the injury to be more serious, British media said.
Carrick is now likely to miss the club's Premier League clash with Arsenal on February 12 and is a doubt for their Champions League last-16 first leg tie against Olympiakos Piraeus 13 days later.
The deep-lying playmaker had only just returned from a six-week layoff following an Achilles tendon problem and his absence is a further blow to embattled manager David Moyes, who is already without injured strikers Wayne Rooney and Robin van Persie.
United are currently seventh in the Premier League, 14 points adrift of leaders Arsenal.
(Reporting by Toby Davis; editing by Patrick Johnston)
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