The Spanish media said that Spartak fans set off the street confrontations, while the Russians accused locals of inciting the unrest.
Spanish police say an officer has died after clashes between police and soccer fans before a Europa League match on Thursday.
The incident once again places a spotlight on Russian hooliganism ahead of this year's World Cup which Russia is hosting.
"La Liga reiterates its strongest condemnation of the violence and continues to work to eradicate it from football", La Liga said in a statement. Five people were arrested - three Russian nationals and two Spaniards.
"As far as security for the 2018 FIFA World Cup is concerned, FIFA has complete trust in the security arrangements and comprehensive security concept developed by the Russian authorities and the Local Organising Committee", a FIFA spokesperson told AFP.
Bilbao lost the match 2-1 on the night, but progressed at the Russians' expense 4-3 on aggregate.
The officer who died was named locally as 51-year-old Inocencio Alonso Garcia who collapsed while intervening in the clashes.
Bilbao had been calm until around 19.45 hours when the most violent of the Spartak "fans" provoked a series of incidents close to the San Mames Stadium throwing bottles, golf balls and fireworks and setting off flares.
The Europa League is UEFA's second tier club competition.
The fighting broke out just before the second leg of the round-of-32 Europa League clash between the two clubs in the streets surrounding San Mamés Stadium in Bilbao, Spain.
Federation Internationale de Football Association said it "deeply regrets" the clashes in Bilbao but had "complete trust" in Russia's ability to ensure a safe World Cup - despite this being just the latest incident of hooligan violence involving Russians.
There was concern ahead of the match because of the reported presence of "Ultra" Russian fans in Bilbao.