The arch-rivals have won 38 league titles between them as well as eight European Cups yet both are struggling in the Premier League this season and are unlikely to relish being pitted against each other at this stage of the competition.
Bookmakers' favourites Borussia Dortmund were also handed a tricky route into the last eight after being drawn against Tottenham Hotspur, who are second in the Premier League behind Leicester City.
Holders Sevilla, chasing a third successive title, will play Basel, while Valencia, who have enjoyed a recent resurgence by winning their last four games under English coach Gary Neville, face an all-Spanish tie against Athletic Bilbao.
Shakhtar Donetsk v Anderlecht
Basel v Sevilla
Villarreal v Bayer Leverkusen
Athletic Bilbao v Valencia
Liverpool v Manchester United
Sparta Prague v Lazio
Borussia Dortmund v Tottenham Hotspur
Fenerbahce v Braga
Two-legged ties to be played March 10 and 17 – Reuters
Original source: United, Reds to clash in Europa League