Martsinkevich -- also known as "Tesak" (Machete) -- last year founded a homophobic group, Occupy Pedophilia, whose aim was to "cure" homosexuals through humiliation and torture.
Several videos of Martsinkevich and his followers humiliating and beating gays circulated on the Internet in 2013.
Last month, a Moscow court sanctioned Martsinkevich's arrest in absentia on extremism charges.
Martsinkevich was detained by Cuban law enforcement agencies in Havana on January 17 for staying in the country without a valid visa for more than 30 days. He arrived in Moscow on January 27.
A spokeswoman for Russia's Investigative Committee, Viktoria Tsyplenkova, said Martsinkevich will be introduced to the charges against him and interrogated on January 28.