Prosecutors at Madrid's High Court want to ask Brazil to extradite Carlos Garcia Julia, who was detained in Sao Paulo, Brazil's biggest city, on December 5th.
But their request still has to be approved by Spain's High Court and then by the Spanish government.
Garcia Julia was among a group of gunmen linked to a far-right organisation who, on January 24, 1977, attacked a law firm working for labour unions.
They killed four Communist lawyers and an administrative staff member in the attack, on Madrid's central Calle Atocha. Garcia Julia was among those convicted in 1980 of the killings.
In 1991, a judge granted Garcia Julia parole and allowed him to move to Paraguay but he disappeared shortly after.
He is believed to have moved around Latin America before finally settling in Brazil where he was reportedly working as an Uber driver using a fake name.
Garcia Julia, who is in his 60s, still has 10 years to serve on his sentence.
The current mayor of Madrid, Manuela Carmena was a lawyer in the office targeted in 1977.