Playing in their maiden Champions League campaign, the Foxes have impressed by winning their group and then overcoming Sevilla 3-2 on aggregate in the last 16.
Their reward for reaching the last eight is a tie against Diego Simeone's high-flying Colchoneros, who have reached the final of the competition twice in the last three seasons.
Leicester's tie is a repeat of a UEFA Cup tie from the 1997-98 season, which the Spanish side won.
The first legs will be played on April 11-12, with the return games on April 18-19, and the final takes place at Cardiff's Principality Stadium on Saturday, June 3.
Full Champions League quarter-final draw:
Atletico Madrid vs. Leicester City
Borussia Dortmund vs. AS Monaco
Bayern Munich vs. Real Madrid
Juventus vs. Barcelona