State radio said the initiative was backed by 40 MPs in the 290-seat assembly. The ISNA news agency put the number at 35.
But it was unclear when, and if, the proposal would be debated and voted upon by the legislature. ISNA said the proposal was for a complete cut in "political" relations.
Speaker Ali Larijani said it was the task of parliament's foreign policy and national security commission to handle the issue of relations with Britain, which is among Western powers accusing Iran of seeking to develop nuclear bombs.
Late last month, Iran summoned the British ambassador in Tehran and Foreign Minister Manouchehr Mottaki threatened Britain with a "slap in the mouth" if it did not stop interfering in Iranian affairs.
That came after Foreign Secretary David Miliband criticised Iranian authorities after eight people were killed in anti-government protests on December 27.
Iranian officials have repeatedly accused Western powers, including Britain, of fomenting street protests that erupted after the Islamic Republic's disputed election in June.
"Considering the sinister actions of the British government towards the Iranian nation, it is the duty of the national security commission to decide about this country … and I thank them for working on it," Larijani said, ISNA reported.
The commission's chairman, Alaeddin Boroujerdi, said the proposal by the MPs had not been coordinated with the commission and called it a hasty move, ISNA reported.
"We should investigate this proposal further and it should be based on our national interest," he said.
Even if the parliament voted to cut or reduce relations with Britain, such a move must be approved by a powerful legislative body, the Guardian Council.
Iran has been convulsed by its most serious domestic unrest since the Islamic revolution in 1979, as protests by opposition supporters against the election result have turned violent. Authorities deny opposition allegations that voting was rigged.
The United States cut relations with Tehran shortly after Iran's 1979 Islamic Revolution that toppled the U.S.-backed shah.
(Reporting by Hossein Jaseb and Ramin Mostafavi; writing by Fredrik Dahl; editing by Samia Nakhoul)
