AC Milan, who sit second in Serie A, are yet to win the competition while United’s only triumph was in 2017.
Arsenal, finalists in 2019, face Greek side Olympiakos, who knocked them out in the last-32 stage last year.
Tottenham will play Croatian side Dinamo Zagreb while Rangers have been drawn against Czech side Slavia Prague, who defeated Leicester City this week.
The eight last-16 ties will be played on 11 and 18 March.
Former Manchester United striker Zlatan Ibrahimovic looks set to return to Old Trafford in the second leg, when the two sides will meet for the 12th time in European competition – all previous meetings having been in the Champions League.
Rangers, who recorded a 9-5 aggregate win over Royal Antwerp on Thursday to reach the last 16 for the second successive season, will take on Slavia Prague for the first time.
Tottenham’s opponents Dinamo Zagreb, who will host the first leg, have lost nine of their last 10 matches against English sides in European competition, winning the other game 2-1 against Arsenal in the Champions League group stage in 2015.
The Gunners last suffered back-to-back eliminations at the hands of the same club in 2014 – against Bayern Munich in the Champions League.
Clubs have until Monday to report any issues surrounding travel to Uefa, after which a decision over confirmed venues will be made.
Europa League last-16 draw
Ajax v BSC Young Boys
Dynamo Kiev v Villarreal
AS Roma v Shakhtar Donetsk
Olympiakos v Arsenal
Dinamo Zagreb v Tottenham
Manchester United v AC Milan
Slavia Prague v Rangers
Granada v Molde