Hirsi Ali, 36, sought asylum in the Netherlands in 1992, claiming to be escaping from an arranged marriage.
She gained Dutch citizenship in 1997 and was elected to parliament in 2003.
A storm erupted about her asylum application last week after a Dutch television documentary interviewed members of her family about her background.
They said she had not been forced into an arranged marriage and had had nothing to fear.
Asked whether she had falsified her asylum application, Hirsi Ali told the programme she lied about her name, age and how she came to the Netherlands.
The programme prompted calls for her to leave parliament and even be stripped of citizenship.
The lawmaker came to international prominence in 2004 when filmmaker Theo van Gogh was murdered after he directed a film she wrote accusing Islam of suppressing women.
Hirsi Ali's office declined to comment on reports in the Dutch media on Monday that she was leaving, but said she planned to address a news conference on Tuesday.
"I think it says something about the state of the country: overly sensitive and provincial"
Jozias van Aartsen
former parliamentary leader, VVD liberals
"I think it is a painful loss both for Dutch society and for politics that Ayaan leaves parliament," said Jozias van Aartsen, a former parliamentary leader of the VVD liberals, which is Hirsi Ali's party and part of the centre-right government.
"I think it says something about the state of the country: overly sensitive and provincial," Van Aartsen told Dutch radio.
The Volkskrant newspaper said Hirsi Ali would start working for the conservative American Enterprise Institute in September after reaching a deal with US authorities about her security.
The paper said Hirsi Ali had also been in negotiations with two other institutes, the Johns Hopkins University and the Brookings Institution, over the past months.
Dutch Radio 1 said the move had been planned for some time, but was brought forward because Hirsi Ali has to leave her heavily-guarded home in August after a court ruled she must move because neighbours had complained of the security risk.
Hirsi Ali went into hiding after Van Gogh's killing due to death threats and only returned to parliament months later.
Rita Verdonk, the immigration minister, vying to be the VVD's lead candidate in the 2007 general election, said on Saturday she would launch an investigation into Hirsi Ali's citizenship.
Hirsi Ali, whose real name is Hirsi Magan, pretended she had come to the Netherlands from Somalia, rather than via Kenya and Germany.
Refugees are usually required to apply for asylum in the first safe country they reach after fleeing.
Immigration has been a hot topic since 2002 when populist Pim Fortuyn said the country could not absorb more foreigners. He was killed by an animal rights activist later that year but mainstream politicians have since taken up his ideas.