Fischer has been wanted for more than 10 years by the United States for violating economic sanctions for playing a chess game in the former Yugoslavia in 1992, the BBC reported.
Japanese officials has said it will not deport the 62-year-old chess player to the United States if Iceland granted him citizenship.
Fisher beat Russia's Boris Spassky for the world chess title in Reykjavik in 1972, and it was a rematch in 1992 in the former Yugoslavia against Spassky that was considered a violation of sanctions by the United States.
Iceland's Parliament voted 40-0 to grant citizenship, with two abstentions.
Icelandic officials said a passport could be ready for Fisher as early as Tuesday.
Yosemite climber falls 30 feet, suffers major injuries
Easer Egg Roll brings thousands to White House