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 at the Nou Camp before the first leg of the cup tie will be played at Bilbao's San Mames stadium on Wednesday, with the return at Barca 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.
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Second-division Mirandes, the only non-La Liga side left in the competition, will play Europa League holders and 2010 Spanish Cup winners Sevilla.
Barca director Javier Bordas told reporters the club were hopeful their appeal against a two-match ban handed to Luis Suarez over a fracas after the first leg of their last 16 tie against Espanyol would succeed and allow the Uruguay striker to feature on Wednesday.
Suarez missed this week's 2-0 victory at Espanyol in the return game, which sent Barca through 6-1 on aggregate.
Barca lost 5-1 to Bilbao in the two-legged Spanish Super Cup in August, the only trophy of six available in 2015 that they failed to win, and Bordas said that proved the Basques were a tough proposition.
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.
(Writing by Iain Rogers, editing by Toby Davis)
