Jones, who can also operate in midfield, was ruled out of the Boxing Day clash at Hull City on Thursday with the problem.
He now looks poised to miss Premier League games against Norwich City and Tottenham and the FA Cup third-round tie with Swansea City in January.
It remains to be seen whether the 21-year-old will be back in time for the first leg of United's League Cup semi-final at Sunderland on January 7, with David Moyes' men set to welcome Swansea to Old Trafford in the league four days later.
The England international had played in all but four of United's 17 league games this season prior to the trip to the KC Stadium.
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