The Spanish giants were drawn Turkish outfit Galatasaray in the final eight of the elite competition, avoiding a potential El Clasico with Barcelona.
Mourinho, who led his side to a
3-2 on aggregate victory over Manchester United in the round of 16, will reunite with Drogba, who played under the Portuguese coach at Chelsea.
The Ivory Coast frontman helped the club that he joined in January edge out Schalke 04 in a thrilling 4-3 aggregate win at the Veltins-Arena on Tuesday.
Meanwhile, Madrid's rivals Barcelona will take on Paris Saint-Germain.
Champions League quarter-final matches:
Malaga vs. Borussia Dortmund
Real Madrid vs. Galatasaray
Paris Saint-Germain vs. Barcelona
Bayern Munich vs. Juventus