Ole Gunnar Solskjaer's side, which holds a 5-0 lead from the first leg of its Round of 16 tie with LASK, could then meet Wolverhampton Wanderers, Olympiacos, Roma or Sevilla in the last four.
Antonio Conte's Inter Milan will take on either Bayer Leverkusen or Rangers in the last eight, if it defeats Getafe in the Round of 16.
Shakhtar Donetsk or Wolfsburg will face Eintracht Frankfurt or Basel in the other quarter-final.
The matches between Getafe and Inter, and Roma and Sevilla, will be held as one-off games at neutral venues in Germany as the first legs could not take place because of the coronavirus pandemic.
All other remaining Round of 16 ties were able to complete their first legs, meaning the reverse fixtures will be held at the home team's stadiums.
Those games will take place over 6-7 August (AEST) before the quarter-finals begin from 11 August (AEST), with those matches being staged in Cologne, Duisburg, Dusseldorf and Gelsenkirchen.
The semi-final venues are yet to be confirmed, but the final on 22 August (AEST) will be held in Cologne.
UEFA Europa League quarter-final draw:
Shakhtar Donetsk or Wolfsburg v Eintracht Frankfurt or Basel
Manchester United or LASK v Istanbul Basaksehir or Copenhagen
Inter or Getafe v Bayer Leverkusen or Rangers
Wolverhampton Wanderers or Olympiacos v Roma or Sevilla
UEFA Europa League semi-final draw:
Winner of QF4 v Winner of QF2
Winner of QF3 v Winner of QF1