Rumours that Fischer, once dubbed the "Mozart of Chess", had been ill had circulated in recent weeks on chess-related websites.
National radio in Iceland, where Fischer had become a resident, reported that he had died after a serious, but unspecified, illness.
Fischer, a child prodigy who once said he liked to watch his opponents squirm, had become an Icelandic citizen after he faced jail in the US for playing the Yugoslavia match with Spassky.
Garry Kasparov, another former world chess champion, hailed Fischer as "the pioneer and the father of professional chess.
"Fischer's chess was so fresh and so new and we all grew up under the strongest impression of Fischer's victories. From an ideological stance it was the fight of an individual against a totalitarian system.
"He had a lot of supporters even in the Soviet Union. No one viewed him as an American fighting Soviets, it was more a great man fighting the mighty machine."
Spassky, who now lives in Paris, was less eloquent on the subject of his old adversary.
Asked by Reuters for his reaction, he said: "It's bad luck for you. Bobby Fischer is dead," then hung up without further comment.
The brilliant and unpredictable American abandoned his world title without moving a pawn by failing to defend his crown in Manila in 1975.
World chess authorities reluctantly awarded it to challenger Anatoly Karpov of the Soviet Union, who was to hold it for the next decade.
Fischer withdrew into himself, not playing in public and living on little more than the magic of his name, although millions of enthusiasts regarded him as the king of chess.
He made headlines when he came out of seclusion to play Spassky in Yugoslavia in 1992, at a time when the country was the target of sanctions during Belgrade's war with breakaway republics.
He vanished after the match, for which he won $3m, and resurfaced after the September 11, 2001, attacks on the US.
In an interview with a Philippine radio station, Fischer praised the strikes and said he wanted to see America "wiped out".
Fischer, who also stirred controversy with anti-Semitic remarks, was granted Icelandic citizenship in March 2005 after eight months in detention in Japan fighting a US deportation order.
Asked who was the greatest player in the world, he once replied: "It's nice to be modest, but it would be stupid if I did not tell the truth. It is Fischer."
Kasparov said that Fischer's withdrawal from chess had been a "great loss" and that he regretted his controversial political statements.
But Fischer's career of "phenomenal victories crushing the best players in the world" was "probably one of the greatest or even the greatest in the history of chess," Kasparov said.