Players can catch Mewtwo in Cerulean Cave in Pokémon Red and Blue after completing the main story.
The cave is near Cerulean City and becomes accessible after the player defeats the Elite Four and the rival. Mewtwo waits deep inside as one of the most powerful and difficult encounters in the original games.
The Pokémon Mansion on Cinnabar Island is important to Mewtwo’s backstory, but it is not the place where the player catches Mewtwo. Research notes there describe Mew and the experiments that led to Mewtwo’s creation.
The same general location returns in later Kanto-based games and remakes, although access requirements can differ. In Pokémon FireRed and LeafGreen, for example, additional post-game progress is required before the cave opens.