The unlikely approval of the visas would mark a startling development in Washington-Tehran ties, following completion of a landmark nuclear deal last year which saw sanctions lifted against the country.
Republicans Mike Pompeo, Lee Zeldin and Frank LoBiondo hand-delivered their applications to the Iranian Interests Section in Washington, and wrote to the Islamic republic's supreme leader Ayatollah Ali Khamenei and the head of Iran's Revolutionary Guards, General Mohammad Ali Jafari, seeking their assistance.
"We look forward to seeing Iranian democracy in action" during the February 26 parliamentary election, the trio wrote in their letter, a copy of which was posted on Pompeo's website.
Tensions have risen within Iran over the elections, with President Hassan Rouhani criticizing moves to exclude thousands of candidates, mostly reformists sidelined from Iranian politics since the disputed 2009 re-election of hardline conservative president Mahmoud Ahmadinejad.
In their letter the US lawmakers sought to "inspect" nuclear sites at Parchin, Fordow and Arak, and requested an briefing by the Revolutionary Guards over the January detention of 10 US Navy sailors.
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The lawmakers also sought "unmonitored" meetings with Americans still held in Iran, and a briefing on recent ballistic missile tests.
No sitting member of Congress is believed to have travelled to the country since the 1979 Islamic revolution, although a former congressman, Democrat Jim Slattery, said he made a pioneering trip there in late 2014.
The lawmakers were more aggressive in their language in the statement accompanying the formal letter.
US President Barack Obama "has consistently rewarded Iran's depraved behavior, providing billions of dollars in sanctions relief to this fanatical regime through implementation of his dangerous nuclear agreement," Pompeo said.
"Given the recent changes brought about by these actions, it is critical that we, as members of Congress, visit Iran and verify whether or not this country will uphold the terms of the nuclear deal."
LoBiondo said approval of the visas would be "a sign of good faith" from Tehran.