The holders Barcelona were drawn alongside Real Sociedad, while Real Madrid's reward for overcoming last season's finalists Sevilla is a two-legged affair with Celta Vigo.
Atletico were drawn alongside Eibar, while Alcorcon - the only non-La Liga side left in the competition - will take on Alaves.
The first legs of the quarter-finals will occur between January 17 and January 19, with the return fixtures to take place one week after.
Barcelona have a record 28 Copa del Rey crowns to their name, while Bilbao, who lost to the Catalan giants in the round of 16, have won the competition on 23 occasions. Real Madrid are third on 19 crowns, while Atletico have lifted the trophy on 10 occasions.
Draw in full:
Alcorcon vs. Alaves Atletico Madrid vs. Eibar Real Madrid vs. Celta Vigo Real Sociedad vs. Barcelona