According to the Icelandic government on Monday, the US ambassador in Reykjavik, James Gadsden, was informed that its offer to Fischer stands.
Washington is seeking to extradite the former chess world champion, a US citizen turned vociferous critic of his country who was detained by Japan in July for allegedly trying to fly out of the country using a revoked US passport.
The 61-year-old is currently being held by Japanese immigration authorities.
Fischer became a national hero in 1972 by winning the "match of the century" against Soviet Grand Master Boris Spassky, in Reykjavik, and wresting the world chess crown from Soviet domination during the Cold War.
Reykjavik responded positively to his residency application last week, arguing that "by not refusing this demand, Iceland is only reacting to its historical ties to the chess player".
Fischer has been sought by the US since 1992 for earning more than $3 million in a chess match staged in the former Yugoslavia in defiance of an international embargo.
But Iceland said it would not withdraw its offer because the US has not officially requested Fischer's extradition from Japan.
"The basis of the decision is that the US government has not asked Japan for the extradition of Bobby Fischer," the foreign ministry said.
Reykjavik maintains that under Icelandic law the statute of limitations on sanction violations from 1992 has expired.
Meanwhile Fischer told Icelandic radio on Monday that he was carrying a valid US passport issued in 1997 when he was arrested in July in Japan. He claimed it was invalidated and destroyed after his arrest.
He said he would die in jail if he were to be extradited to the US. "There is no question, that I will never get out alive," he said.