Division over the future of ties with the US centre on concerns among regime conservatives that moderate politicians are trying to ease tensions with Washington through co-operation over the conflict in Syria.
Hamid-Reza Taraghi, a conservative politician, said on Monday that the "unjustifiable" handshake was "probably accidental" but that parliament would determine whether Mr Zarif's gesture was deliberate or part of a "conspiracy". If deliberate, Mr Zarif would "definitely face impeachment".
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"We don't think [Mr Zarif] could dare to act contrary to the orders of the supreme leader," Mr Taraghi said.
Mansour Haghighatpour, another political conservative and member of the parliament's national security and foreign policy committee, said parliament would not remain indifferent should it be confirmed that Mr Zarif had ignored the regime's red lines.
"There should not be talk of any handshakes with US officials as long as the US supports terrorists and does not respect our nation's rights," Mr Haghighatpour told Tasnim, a news agency close to the Revolutionary Guards.
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Whether the threats of impeachment would be acted on remains unclear. Mr Zarif is seen by many Iranians as a national hero who spearheaded nuclear talks with the world's major powers to achieve a breakthrough deal that would lift crippling economic sanctions.
In a recent television interview in the US, Hassan Rouhani, the centrist president, said that despite "the distance, the disagreements, the lack of trust" between Tehran and Washington, both countries were moving towards decreasing the enmity between them. He also indicated in the US the possibility of co-operation with Washington over regional issues.
"The handshake has symbolic importance in line with what has happened during the nuclear talks to extend co-operation from nuclear matters to regional issues," said a reform-minded analyst. "The issue of the US is linked to domestic politics."