The Blues will host the La Liga leaders in the first leg at Stamford Bridge, in a match that is likely to be played behind closed doors after owner Roman Abramovich was sanctioned by the UK and EU.
The defending European champions will then face a trip to the Bernabeu in the return leg one week later, with the winner of the tie facing either Manchester City or Atletico Madrid in the semi-finals.
Reacting to the draw ahead of his side’s trip to Middlesbrough this weekend, Chelsea manager Thomas Tuchel said: “It’s a tough one. A challenge cannot be much higher than playing in the Bernabeu with spectators. There is also big excitement around this fixture.”
Liverpool will play Benfica with the winner of that quarter-final facing either Villarreal or Bayern Munich in the other side of the draw.
Chelsea suffered a second-leg scare in their last-16 clash against Lille when Burak Yilmaz scored a penalty for the French champions but goals from Christian Pulisic and Cesar Azpilicueta gave the Blues a 2-1 victory on the night and a comfortable 4-1 triumph on aggregate.
Full Champions League quarter-final draw:
Chelsea vs Real Madrid
Manchester City vs Atletico Madrid
Villarreal vs Bayern Munich
Benfica vs Liverpool
Full Champions League semi-final draw:
Manchester City / Atletico Madrid vs Chelsea / Real Madrid
Benfica / Liverpool vs Villarreal / Bayern Munich
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