The stage of the competition features eight runners-up from their Europa League groups, which includes United, playing the eight teams who were third in their Champions League group.
Former United manager Jose Mourinho’s Roma side will play FC Salzburg, while Juventus face Nantes.
The first legs are on 16 February, with the second legs on 23 February.
The winners go into the last-16, which includes Arsenal after they topped their Europa League group.
United, who finished second behind Real Sociedad in their group, will be away for the first leg at the Nou Camp, followed by a meeting at Old Trafford seven days later.
Erik ten Hag’s side were heavily linked with Barcelona midfielder Frenkie de Jong during the summer, but the Dutch international ended up staying with the Spanish side.
Barcelona, who are top of La Liga, finished third behind Bayern Munich and Inter Milan in their Champions League group.
“We will have to compete, going to Manchester against a really top rival and an historic side which has grown a lot with the arrival of [Erik] ten Hag,” said Barcelona boss Xavi.
“[They have] great individual players and it is the toughest rival once again in the Europa League.
“It’s a massive challenge for us.”
Sevilla have won the Europa League six times and will take on Dutch side PSV Eindhoven.
Full play-off round draw
Juventus v Nantes
Sporting Lisbon v Midtjylland
Shakhtar Donetsk v Rennes
Ajax v Union Berlin
Bayer Leverkusen v Monaco
Sevilla v PSV Eindhoven
FC Salzburg v Roma