Garcia is stepping up his inquiry into voting procedures for the 2018 and 2022 World Cups. He plans to visit every country directly involved in the voting for the finals awarded to Russia for 2018 and Qatar for 2022.
But Garcia, a former US federal prosecutor, is on a blacklist Moscow issued in June after Washington named 18 Russians barred from the United States for alleged involvement in lawyer Sergei Magnitsky's death or other gross rights abuses.
Garcia is barred from Russia because of his role in the prosecution of Viktor Bout, a Russian who was long wanted by US authorities on suspicion of arms trafficking and is now serving a 25-year prison term in the United States.
Russia's "Guantanamo List" includes Americans it accuses of being involved in torture at prisons and those involved in what it says have been the unfair, politically motivated arrests of Bout and other Russians by US authorities.
"Let there be no doubt: We intend to react firmly to unfriendly attacks and unceremonious infringements on the rights of Russian citizens," the Foreign Ministry said.
"Anyone who is involved in such things should think hard."
The ministry said the United States would be to blame for any consequences of Garcia's inability to enter Russia.
"Questions arising in connection with visa blacklists should be addressed to the initiators and executors of the extraterritorial and discriminatory 'Magnitsky Act', which contradicts norms of international law," it said.