We will play the first leg at home on Tuesday 9 April, kick-off 8pm, with the second leg to be played away on Wednesday 17 April, kick-off 8pm.
The five-time Balon d'Or victor also drew a blank in prior quarter-final clashes with Juventus, twice against Atletico Madrid and in Barca's 2015 tie with PSG.
Liverpool will go top of the Premier League on Sunday if they extend their winning streak against Fulham.
Former Liverpool goalkeeper Chris Kirkland was delighted by the draw, telling Sky Sports News: "I'd prefer being at home for the second leg but I'm extremely happy".
However, things could get a little tight around the midfield as Naby Keita faces a late fitness test in order to prove that he's recovered from the niggle that kept him out of the Bayern Munich game.
Man United or Barcelona vs.
Four English teams reached the last eight for the first time in 10 years, though only one national derby was drawn.
"But I think the fans would go with the league, they want to retain that - they still haven't fallen in love with the Champions League because of the issues with Financial Fair Play and the amount of players you are allowed in the squad". The two teams meet again in the league just days after the second leg, on April 20.
Solskjaer's 93rd-minute goal at Camp Nou in 1999 won the Champions League final for United, securing a 2-1 victory over Bayern Munich.
The Reds have won each of the last five meetings with the Cottagers across all competitions - a new club record run versus the London side. There are six teams in the Premier League and we can not say they are bad because all of them are in quarter-finals [Arsenal and Chelsea are in the last eight of the Europa League].
Cristiano Ronaldo's hat-trick against Atletico Madrid kept Juventus' European dream alive.
Expectations are high that Guardiola, who twice guided Barcelona to Europe's top prize, will lead the Premier League champions into the final four for the first time since taking charge in 2016.