The last eight also produced an all English pairing as Premier League leaders Manchester City were drawn against Liverpool, the only team to have beaten Pep Guardiola's side in the league this season.
In the other ties, Barcelona will face AS Roma, bringing back unhappy memories for the Italian side who lost 6-1 on their last visit to the Nou Camp in November 2015, and Sevilla, who beat Manchester United in the last round, will take on Bayern Munich.
The first legs will be played on April 3/4 with the return matches one week later.
Juventus, who will host the first leg, were thumped 4-1 by Real in last season's final in Cardiff, although the Turin side came out on top when they last met over two legs, in the semi-finals in 2015.
Overall, the teams have faced each other 19 times with eight wins for Juve, nine for Real -- including the 1998 final -- and two draws.
"In last year's final we only played well for the first 45 minutes," said Juventus vice-president and former midfielder Pavel Nedved.
"However, over two legs, I think we will have a better chance... they will be two very tough contests, but we will try to play the Juventus way."
Liverpool's tie against Manchester City will be the first meeting between English sides in the quarter-finals since 2010/11 when Chelsea faced Manchester United.
City have won only one of their last eight meetings with Liverpool in all competitions, if you exclude their penalty shootout triumph in the 2016 League Cup final.
"It’s always common in a draw that you’ll get your neighbour, more or less, but to be honest I don’t mind," said Liverpool manager Juergen Klopp.
"I don’t think they thought before the best draw they could have got is Liverpool. That’s a sign for us and how strong we can be... Thank God it’s football and nothing is decided."
Former Roma stalwart Francesco Totti said his former team, who are considered the outsiders alongside Sevilla, should not be worried about facing Barca.
"Football is beautiful because anything can happen, especially in these type of games," he said. "We've already come a long way, finishing ahead of Chelsea and Atletico Madrid in the group stage and knocking out Shakhtar Donetsk."
Bayern, the runaway Bundesliga leaders, promised they would not underestimate Sevilla.
"Sevilla are not an easy team. They eliminated Manchester United," said Bayern sports director Hasan Salihamidzic.
"It certainly is no easy draw, they did not come through to this stage by accident," added midfielder David Alaba.
(Additional reporting by Karolos Grohmann, Alan Baldin and Cecile Mantonvani; Writing by Brian Homewood; Editing by Jon Boyle and Toby Davis)
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