The Serie A side, beaten 4-1 in Cardiff last season by Real, will host the first leg in Turin following the draw at UEFA headquarters in Nyon.
The last eight also produced an all-English pairing as Liverpool were thrown together with English Premier League leaders Manchester City.
In the other ties, FC Barcelona will face AS Roma, bringing back unhappy memories for the Italian side who lost 6-1 on their last visit to Camp Nou.
Sevilla will take on Bayern Munich.
Juventus are not completely ruled out of the tie, as 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 Juventus, nine for Real — including the 1998 final — and two draws.
Liverpool’s tie against Manchester City will be the first meeting between English sides in the quarter-finals since 2010/2011 when Chelsea faced Manchester United.
Manchester 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 English League Cup final.
Former Roma stalwart, Francesco Totti, said his former team, who are considered the outsiders alongside Sevilla, should not be worried about facing FC Barcelona.
“Football is beautiful because anything can happen, especially in these types of games.
“We’ve already come a long way, finishing ahead of Chelsea and Atletico Madrid in the group stage and knocking out Shakhtar Donetsk,” he said.
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.
The first legs will be played on April 3 and April 4, with the return matches one week later.
The Full Champions League Quarter-final Draw
FC Barcelona vs Roma
Sevilla vs Bayern Munich
Juventus vs Real Madrid
Liverpool vs Manchester City