United knocked out AC Milan to reach the quarter-final stage and have now been paired with Granada, who have eliminated Napoli and Molde during the knockout phase.
Arsenal, meanwhile, saw off Olympiacos in the round of 16 and will now meet Slavia Prague, who have knocked out Leicester City and Rangers.
In the other two quarter-final ties, Ajax will play Roma, while Dinamo Zagreb, who completed a dramatic upset to knock out Tottenham on Thursday evening, will meet Villarreal.
The Europa League semi-final draw was also made on Friday and if United beat Granada, they will meet either Ajax or Roma.
Arsenal, if they progress in their tie with Slavia Prague, will play either Dinamo Zagreb or Villarreal, who are currently managed by their former boss Unai Emery.
EUROPA LEAGUE QUARTER-FINAL DRAW
Granada vs Manchester United
Arsenal vs Slavia Prague
Ajax vs Roma
Dinamo Zagreb vs Villarreal
EUROPA LEAGUE SEMI-FINAL DRAW
Granada or Man United vs Ajax or Roma
Dinamo Zagreb or Villarreal vs Arsenal or Slavia Prague
The first leg of the quarter-final ties are due to take place on April 8, while the return legs will be played on April 15.
The first leg of the semi-final fixtures will be held on on April 29, while the second leg will take place on May 6.
The final, which will be held at the Gdansk Arena in Poland, is scheduled for May 26.