Rojo dislocated his left shoulder after a fall in training - the same injury he suffered in a 1-0 defeat at Manchester City last season.
It is not known how long the 25-year-old will be on the sidelines, but British media said the defender was expected to be ruled out for four weeks.
Rojo joins a lengthy list of defensive absentees for Louis van Gaal's side, including Phil Jones and Luke Shaw.
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Centre back Chris Smalling (groin) and full back Matteo Darmian (hamstring) are doubts for Saturday's Premier League trip to Bournemouth after picking up knocks in Tuesday's 3-2 Champions League loss at VfL Wolfsburg.
That defeat saw United knocked out of Europe's elite competition and relegated to the second-tier Europa League.
(Reporting by Shravanth Vijayakumar in Bengaluru; editing by Ken Ferris)
