The Red Devils will host the Catalan giants at home in the first leg.
The draw was made the other way round, but was reversed to avoid United and Manchester City both playing their home legs at the same.
The two sides have played each other 11 times in the past, with United winning three, losing four and drawing four.
Those past meetings included the 2009 and 2011 Champions League finals, which were both won by Barcelona.
The 2011 final was the recent competitive meeting between the clubs.
Ole Gunnar Solskjaer’s side recorded a remarkable comeback win over Paris Saint-Germain in their last-16 tie, winning 1-3 at Parc des Princes in a memorable second leg.
Barca recorded a comfortable win over Lyon in their last-16 tie.
The winners of this tie will face the winners of the Liverpool vs Porto tie, raising the prospect of a Manchester United vs Liverpool Champions League semi-final.
Champions League quarterfinals draw.
Ajax vs Juventus
Liverpool vs Porto
Tottenham vs Man City
Man Utd vs Barcelona.