Arsenal, hoping to qualify for the 17th consecutive season, must face Turkey's Besiktas and twice former champions Porto were also handed a difficult draw as they were paired against Lille, with the French club at home the first leg.
Rafael Benitez's Napoli finished third in Serie A last season, only enough for a place in the playoff round after Italy slipped down the national coefficients which determine how many places each league gets in the tournament.
"Among the five possible opponents, with Athletic we have been landed with the most difficult," Benitez said.
"They're a team who play good football and, at home, the 50,000 at San Mames give them a big push. It will be a very good, and very intense game."
Napoli also won the Coppa Italia and were eliminated in the group stage of the Champions League in Benitez's first season in charge.
The performance was regarded as no more than satisfactory and he will be expected to improve on that showing this term with one of the most expensive squads in Serie A.
Bilbao, fourth in La Liga, were beaten Europa League finalists in 2012.
If Napoli fail to beat Bilbao, Italy will be left with only two representatives, Juventus and AS Roma, in the group stage.
TURKISH OPPOSITION
Arsenal also met Turkish opposition at the same stage last season when they beat Fenerbahce 5-0 on aggregate over two legs.
"It's nice we have got the second leg at home," Arsenal club secretary David Miles told reporters.
"They (the playoff ties) come relatively early in the season, and our three German World Cup winners won't even be back, or will be just starting training then, so probably we won't have the full squad to choose from,"Nevertheless, we've got a good squad, quality in depth and I'm sure will be okay."
The first legs will be played on Aug. 19/20 with the returns on Aug. 26/27.
Scottish champions Celtic, reprieved after third qualifying round opponents Legia Warsaw fielded an ineligible player in their 6-1 aggregate win, will meet Slovenian champions Maribor.
Former European champions Steaua Bucharest were paired with Bulgaria's Ludogorets Razgrad while Cypriot champions APOEL Nicosia, who reached the quarter-finals two seasons ago, were drawn against Danish counterparts AaB Aalborg.
Malmo, runners-up in 1979, play Austrian champions Salzburg, who are attempting to reach the group stage for the first time since Red Bull's takeover in 2005.
FC Copenhagen will meet Bundesliga side Bayer Leverkusen, Belgium's Standard Liege face Zenit St Petersburg and Slovan Bratislava play Belarusian champions BATE Borisov in the other of the 10 ties.
UEFA said that the vanishing spray, used at the World Cup to display the correct distance between a free kick and the defensive wall, would be used in the tournament, starting with the playoff round.
(Editing by Ed Osmond)
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