The Spanish media said that Spartak fans set off the street confrontations, while the Russians accused locals of inciting the unrest.
Spartak won the second leg of the round of 32 tie 2-1 but were knocked out of Europe's second-tier club competition 4-3 on aggregate after losing the first leg 3-1 in Moscow.
The deadly clashes in Bilbao raise concerns over football violence four months before Russian Federation is due to host the World Cup.
European football's governing body Uefa said it "strongly condemned the violent clashes". Five people were arrested - three Russian nationals and two Spaniards.
Russian local authorities are putting strict security procedures in place to counter hooliganism during the world cup tournament scheduled for June.
Marca reports that the man who died, Inocencio Arias Garcia, 50, suffered a heart attack during the clashes outside of San Mames Stadium with ultras and later died at the hospital.
REUTERS FOOTBALL Spartak Moscow's Luiz Adriano in action with Athletic Bilbao's Markel Susaeta
"The Russian ultras shouldn't have traveled to Bilbao due to their history", Javier Tebas told the sports daily As.
A column on the website of Russia's main sport channel Match TV accused the police of "having insufficient experience" in dealing with converging groups of fans.
The fight broke out in the streets surrounding the Basque club's San Mames stadium, just hours to their second leg meeting, sadly claiming the life of a policeman. "Authorities should make lists of radical fans to make sure these people can't get visas and travel overseas to see away games".
There was concern ahead of the match because of the reported presence of "Ultra" Russian fans in Bilbao, and a large police force was deployed to try to prevent fan trouble.
Six years ago, a fan died in Bilbao in clashes after a match between Athletic and German club Schalke.
Earlier this week, a Russian fan accused causing serious injuries to an English supporter who was left in a coma after rioting at Euro 2016 was arrested at a German airport while in transit en route to Bilbao to watch Thursday's fixture.
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