Real Madrid will face Bayern Munich in the Champions League quarterfinals, while Barcelona take on Juventus.
Premier League champions Leicester City, who secured a surprise success over Sevilla in the round of 16, will play Atletico Madrid.
The final tie sees Borussia Dortmund play Monaco, who defeated Manchester City on away goals in the last round.
Real Madrid's director of institutional relations, Emilio Butragueno, was disappointed at drawing Bayern Munich and said he would have hoped to face them later in the competition.
"Bayern are one of the worst opponents we could get, no doubt," Butragueno told beIN Sports. "But [it is disappointing] also for them. We know each other very well, the two teams.
"We must go into the games knowing how difficult it will be. They have top-class players in every position, with so much experience. It is a great club. We have a very good relationship with them. We would have liked to meet them later in the competition, but the draw is this, and we must take it on."
The second leg at home at the Bernabeu was a slight advantage, Butragueno said, assuming Madrid get a good result in Germany.
"If we get a good result there, then yes," he said. "The second game at home gives you a very slight advantage. But we must play a good game there. We know that 20 good minutes from Bayern can be deadly for us. Today is not the best draw we could have got. But we are a team with top players and lots of experience. It is a very nice tie to be able to play, and everybody will be watching these games."
The tie brings current Bayern coach Carlo Ancelotti back to the club where he won the Champions League in 2013-14, his first season in charge. But he was then sacked by president Florentino Perez after he finished his second season without winning a major trophy.
"Carlo is a great coach, with so much experience," Butragueno said. "He has won everything as a player and as a coach, which tells you everything. We will talk these days. We get on very well with him. He is a gentleman who knows a lot about football. So it is not such good news for us."
Barcelona vice president Jordi Mestre hopes to avoid the mistakes made against Paris Saint-Germain in the round of 16 in order to ensure a smoother passage against Juventus.
"The draw is the draw. There are no easy teams in the quarterfinals, but we always get tough ties," he told beIN Sports. "There's a good dynamic at the moment, and morale is sky high, but that doesn't mean we're going to go through.
"I respect Juve a lot. The advantage of playing the second leg at home is conditioned a lot by what happens in the first game -- we hope it's not the same as Paris. We have to score in Turin."
Atletico director Clemente Villaverde was asked on beIN Sports whether his team were delighted to have drawn Leicester, seen by most pundits in Spain as the weakest of the eight teams left in the competition.
"I don't know about that, as in their last tie and in the groups they showed they are a team who must be taken into account," Villaverde said. "When you are in the quarterfinals, among the best eight teams in Europe, you must respect all opponents.
"Leicester are here on their own merits. They overcome a very important team like Sevilla in the last round. For sure it will be a very difficult tie. This will be really difficult for us. We played them in the UEFA Cup in the 1990s, and we hope to follow the same steps this time."
Villaverde added that Atletico would not make the same mistakes as Sevilla did in the round of 16 against Leicester, with Diego Simeone's team having the experience to not let chances slip in such big games.
"Atletico must learn to take their chances when they get them," he said. "Sevilla had quite bad luck over the two games. They were better over the two games. But football is like that. If you have chances and cannot kill off the game, there is a chance the other team can come back.
"We have shown we can compete in this competition in these last years, and we are confident we can avoid [such] mistakes and ensure our passage."
Playing the first leg at home was not an issue for Atletico, Villaverde said.
"Usually we are not too worried about the order of the games," he said. "In ties like this, each game will be totally different. We need to know how to compete in each game, depending on how the game develops. It is not relevant if you play home or away first."
Ties to be played April 11 and 12; second legs on April 18 and 19:
Atletico Madrid v Leicester City
Borussia Dortmund v Monaco
Bayern Munich v Real Madrid
Juventus v Barcelona
Borussia Dortmund chief executive Hans-Joachim Watzke was happy to avoid Bundesliga rivals Bayern Munich but said they will not take Monaco, the highest scorers in Europe's top five leagues, lightly.
"We wanted to avoid Bayern, and thus the draw is good," Watzke told Sky Deutschland. "But we've seen the games against Manchester City, and their goal difference is at plus 58. Still, we can't complain. Overall, it's a nice draw.
"Certainly, their stadium is not a lion's den, but they've turned the tie against City around. At this level, it does not matter whether you play the first leg at home or away. Sure, you prefer playing the second leg at home, but we take things as they come."
Monaco vice-president Vadim Vasilyev called on his club's fans to match the world-renowned passion of the Dortmund supporters.
Monaco's average attendance for a Ligue 1 game is just over 9,000, while nearly 25,000 cram into Dortmund's famous Sudtribune stand alone virtually every match.
"It's a great game, because Dortmund are a team that plays attacking and attractive football," Vasilyev told beIN Sports.
"I have heard a lot of talk about fans. Against Manchester City, the atmosphere was exceptional at the Louis II. I hope our fans will again turn out in numbers for the quarterfinal.
"For me, the Champions League is a bonus. The priority remains the league. We have nine finals to play. This team doesn't set itself limits."
The semifinal draw is on April 21, with the semifinals themselves coming on May 2-3 and 9-10.
This year's final is on June 3 in Cardiff, Wales.