Catalans face Ligue 1 leaders, while Real Madrid will take on Galatasaray.
Champions League favourites Barcelona will play big-spending Paris Saint-Germain in the pick of the quarter-final matches after the draw was held in Nyon, Switzerland on Friday.
Former England captain David Beckham will get the chance to return to the Nou Camp where he famously helped Manchester United defeat Bayern Munich 2-1 in the 1999 final although they will be without leading scorer Zlatan Ibrahimovic who is suspended for the first-leg at the Parc des Princes.
Lionel Messi will lead four-time champions Barcelona after the Argentine superstar scored twice and inspired the 4-0 thrashing of AC Milan on Tuesday that overturned a 2-0 deficit.
“It’s the toughest draw we could get and it’s going to be difficult,” said PSG’s sporting director and former player Leonardo.
“It’s not going to be easy but the two matches will bring us a lot of enjoyment and they will be great matches of football,” added the Brazilian.
Last year’s losing finalists Bayern Munich, will play the first-leg at home against Italian giants Juventus, while record nine-time champions Real Madrid will take on a Galatasaray team attempting to become the first side from Turkey to lift the trophy.
“Galatasaray are a great team and have fantastic fans so we are expecting two tough matches,” said former Real Madrid striker Emilio Butragueno.
“They have some great players like (Wesley) Sneijder and (Didier) Drogba but our objective is to win that 10th European Cup. That is our goal,” added the former Spanish international.
In the final quarter-final, Spanish outsiders Malaga were drawn against 1997 champions Borussia Dortmund with the first-leg to be played in Spain.
The final is set for Wembley on May 25 although there are no English clubs remaining in the last eight for the first time since 1996, while defending champions Chelsea failed to make it out of the group stages.
Champions League quarter-final draw:
Matches to be played on April 2/3 and April 9/10
Paris Saint-Germain (FRA) v Barcelona (ESP)
Malaga (ESP) v Borussia Dortmund (GER)
Real Madrid (ESP) v Galatasary (TUR)
Bayern Munich (GER) v Juventus (ITA)