The officer died in hospital in the northern Spanish city of Bilbao. He had been taken there when the regional police force was trying to stop street battles between supporters of Spartak and Athletic before their Europa League last 32, second leg at the San Mames stadium.
Nine people were arrested following the clashes. Spanish newspaper El Mundo said two Russian ultras were taken to hospital with injuries, one with stab wounds.
The World Cup begins on June 14 in Russia and local authorities are putting strict security procedures in place to control hooligans, including Russian fans who caused trouble at the 2016 European Championship in France and other soccer events.
"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 Reuters via email.
"As demonstrated during the FIFA Confederations Cup last year, Russia's already high security standards have been adapted to meet the specific needs of such major sporting events.
"Naturally, FIFA is also in constant contact with all stakeholders regarding the ongoing risk assessment, which enables the implementation of overall preventive measures in cooperation with national and international enforcement authorities."
Spartak won the match 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.
Russian supporters will return to Spain next month, however, after Lokomotiv Moscow were paired with Atletico Madrid in Friday's last-16 draw. They visit the Spanish capital in the first leg on March 8.
Two other Russian sides remain in the tournament: CSKA Moscow, who were paired with French side Olympique Lyonnais and Zenit St Petersburg, who face German team RB Leipzig.
Josu Zubiaga, vice-chairman of security for the Basque government told Spanish radio station Cadena Ser that the policeman, named as Inocencio Alonso, 51, had not died as a result of violent contact.
Roberto Seijo, general secretary for trade union ErNE of the Basque police force, however, described the death as a direct result of the violence.
"The death of our colleague did not take place while sitting in an armchair but as the consequence of these incidents," he said, quoted in El Mundo.
(Reporting by Richard Martin; Editing by Christian Radnedge)
