The draw for the last eight kept the two remaining Premier League giants apart.
United claimed a 1-0 win at AC Milan on Friday (AEDT) to advance 2-1 on aggregate and its reward is an apparently kind draw against LaLiga's Granada, a winner over Molde in the Round of 16.
If United advances, it could meet Ajax, which is defeated in the 2016-2017 final. The Dutch side faces Roma for the right to play the Red Devils or Granada.
Ajax was too strong for Young Boys, while Roma dumped out Shakhtar Donetsk in the Round of 16.
Arsenal, which knocked out Olympiacos despite losing at home, will be wary of its quarter-final opponent Slavia Prague.
Slavia has already eliminated Leicester City and Rangers, and has now been paired with the 2018-2019 finalist, visiting Emirates Stadium before hosting the reverse fixture.
Arsenal's half of the draw is completed by Dinamo Zagreb and Villarreal.
Dinamo defeated Tottenham Hotspur in a stunning second-leg comeback on Friday (AEDT) and could get the chance to complete a north London double if it meets the Gunners.
A meeting between Arsenal and Villarreal would also have significance, with the Spanish outfit coached by Unai Emery, Mikel Arteta's predecessor and a three-time UEFA Europa League winner with Sevilla.
Emery's men swept aside Dynamo Kiev in the Round of 16.
UEFA Europa League quarter-final draw:
Granada v Manchester United
Arsenal v Slavia Prague
Ajax v Roma
Dinamo Zagreb v Villarreal
Winner of QF 1 v Winner of QF 3
Winner of QF 4 v Winner of QF 2