The French international is also ruled out of the World Cup qualifier against Luxembourg on March 25, followed by a home friendly against Spain three days later.
"I don't know how long (Pogba will be out) for sure," Mourinho told British media. "But there will be no Middlesbrough and no national team."
Defender Daley Blind also left the pitch after a bump on his head and is doubtful for the next game.
Seventh-placed United will also be without suspended striker Zlatan Ibrahimovic and midfielder Ander Herrera for their trip to the Riverside.
Middlesbrough, who are second-bottom of the standings and without a win in 10 matches, sacked manager Aitor Karanka on Thursday.
(Reporting by Hardik Vyas in Bengaluru; editing by Sudipto Ganguly)
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