Should they advance, Erik ten Hag’s side will face either Juventus – whose squad includes former United midfielder Paul Pogba – or Arsenal’s conquerors Sporting Lisbon in the semi-finals.
In the other last-eight ties, Roma face Feyenoord and Bayer Leverkusen meet Union Saint-Gilloise.
It will be the first meeting between the sides since the 1964-65 Cup Winners’ Cup, which West Ham went on to win.
If they defeat their Belgian opponents, West Ham face AZ Alkmaar or Anderlecht in the semi-finals.
Lech Poznan take on Fiorentina and Basel play Nice in the other ties.
The first legs of of both competitions take place on 13 April, with the return games on 20 April.
The semi-finals are scheduled to be played on 11 and 18 May.
Their squad includes Brazil defender Alex Telles, who is on loan from United.
“We have to be aware of it.”
Ten Hag’s side, who won last month’s Carabao Cup, are also still in the FA Cup and host Fulham in the quarter-finals of that competition on Sunday (16:30 GMT).
Europa League quarter-final draw
Manchester United v Sevilla
Juventus v Sporting Lisbon
Bayer Leverkusen v Union Saint-Gilloise
Feyenoord v Roma
Europa League semi-final draw
Juventus or Sporting Lisbon v Manchester United or Sevilla
Feyenoord or Roma v Bayer Leverkusen or Union Saint-Gilloise
Europa Conference League quarter-final draw
Lech Poznan v Fiorentina
AZ Alkmaar v Anderlecht
Basel v Nice
Europa Conference League semi-final draw
Lech Poznan or Fiorentina v Basel or Nice