Barcelona failed to get out of a group which included Bayern Munich and Inter Milan, dropping into the Europa League for the second season running. United, meanwhile, finished second in their group behind LaLiga side Real Sociedad.
The tie will be played over two legs on Feb. 16 and Feb. 23, with the first leg taking place at Camp Nou and the return game at Old Trafford a week later.
It is a draw both sides would have liked to avoid, with Barcelona and United among the favourites to go all the way in the competition.
Barcelona are top of LaLiga having lost just one game in the league this season following summer spending of over €150 million, including the €45m signing of Robert Lewandowski.
United, meanwhile, have shown positive signs under new coach Erik ten Hag this season, although they saw a nine-game unbeaten run ended by Aston Villa at the weekend.
"The draw has given us the most difficult opponent once again," Barca coach Xavi said. "But we will look forward to it and we will compete.They are a historic side, they have grown a lot with the arrival of Ten Hag, they have great players. It's the worst draw we could have got. Once again.
"No luck. Sometimes it's a plus to play against the best sides. Maybe it will bring the best out of us here and at Old Trafford. It's a challenge."
It will be the first meeting between the two clubs since they faced each other in the Champions League quarterfinal in 2018-19. Barcelona won both games on that occasion, progressing 4-0 on aggregate.
The two sides also met in the 2009 and 2011 Champions League finals, with Barcelona winning both times, while United's last win against the Spanish side came in the 2008 Champions League semifinal.
"We are going to try to win it. The club has never won the Europa League and we're going to go all out to try and win it, there is nothing else for it," Xavi said.
"Cristiano is a great player. He has marked an era in football and he can still make the difference in games. The have great players, great individuals, but I would highlight the group.
"We will analyse them thoroughly. They're a great team, not just because of Cristiano. They are a historic club and they also have the necessity to win."
Six-time winners of the competition Sevilla will meet Dutch league leaders PSV Eindhoven, while Ajax will take on Union Berlin, who after setting the early pace in the Bundesliga have now lost two of their last three matches.
The Europa League playoff round pairs the eight teams that finished second in the group stage against the eight sides that finished third in their Champions League group, thus dropping down into Europe's second tier competition.
This season's final will be played at the Puskas Arena in Budapest on May 31, with Budapest-based Ferencvaros among the other teams that will enter the competition in the next round.
Europa League playoff draw:
Barcelona vs. Manchester United
Juventus vs. Nantes
Sporting vs. Midtjylland
Shakhtar Donetsk vs. Rennes
Ajax vs. Union Berlin
Bayer Leverkusen vs. Monaco
Sevilla vs. PSV Eindhoven
FC Salzburg vs. Roma