Barca dropped to the Europa League for the second straight season after being eliminated from the Champions League group stage, while United finished second in their Europa League group.
The two teams last faced off in the 2018–19 Champions League quarterfinals, with Barca winning 4-0 on the road.
The eight Europa League group winners all automatically qualify for the last 16, where the winners of the knockout phase play-off matches will advance.
Juventus was paired with French side Nantes in a match that will be a rematch of the Italians’ victory in the 1996 Champions League semi-final after Nantes had also been eliminated from Europe’s premier club competition.
Monaco will play Bayer Leverkusen, and Jose Mourinho’s Roma will play Red Bull Salzburg.
Sevilla, who has a record six victories in the Europa League or its precursor, the UEFA Cup, will take on PSV Eindhoven.
Ajax was paired with Bundesliga’s top-scoring Union Berlin, and Shakhtar Donetsk will face Rennes of France.
Along with Fenerbahce, Real Betis, Union Saint-Gilloise of Belgium, Real Sociedad, Feyenoord, Freiburg, and Ferencvaros, Premier League leaders Arsenal have already advanced to the round of 16.
The Europa League final game for this season will take place on May 31 in Budapest.