The Washington Post says Iranian officials have used intermediaries to make the request, citing US officials, Iranian analysts and foreign diplomats.
The requests follow a May 8 letter from Iran's hardline leader Mahmood Ahmadinejad to US President George W. Bush, the first such communication between an Iranian and US leader in more than 25 years.
Senior Iranian officials have asked "a multitude of intermediaries" to send the word to Washington that it is open for direct talks, the Post reported, citing Saeed Laylaz, a former Iranian government official and prominent analyst in Tehran, and several diplomats.
Conduits for the messages include the head of the International Atomic Energy Agency, Mohamed ElBaradei - in Washington for talks with Secretary of State Condoleezza Rice - and national security adviser Stephen Hadley, the Post reported.
Other conduits have included UN Secretary General Kofi Annan and officials in Indonesia, Kuwait.
"They've been desperate to do it," a European diplomat in Tehran told the Post.
According to the daily, Mr Ahmadinejad's letter was also implicitly endorsed by Iran's supreme leader, Ayatollah Ali Khamenei.
"You know, two months ago nobody would believe that Mr. Khamenei and Mr. Ahmadinejad together would be trying to get George W. Bush to begin negotiations," Mr Laylaz told the Post in Tehran.
"This is a sign of changing strategy. They realise the situation is dangerous and they should not waste time, that they should reach out."
Washington has not had diplomatic relations with Iran since April 1980, following the seizure of the US embassy in Tehran in 1979 in which 52 Americans were held for 444 days.