Friday's draw pitted the record winners, who lost to Real Madrid in the 2011 and 2014 finals, against the Basque side, whose 23 Cup victories, their most recent coming in 1984 against Barca, are only bettered by the Catalan club's 27.
The sides face each other in La Liga on Sunday before the first leg of the cup tie will be played at Bilbao's San Mames stadium on Wednesday, with the return at the Nou Camp a week later.
Atletico Madrid, the 2013 winners, were drawn to play Celta Vigo, who have never won the competition, while Gary Neville's Valencia, whose most recent and seventh win came in 2008, will meet Las Palmas.
Second-division Mirandes, the only non-La Liga side left in the competition, will play Europa League holders and 2010 Spanish Cup winners Sevilla.
The semi-finals will be played over two legs at the start of February and the final on May 21 at a yet-to-be-determined venue.