Donald Trump has responded angrily to Iranian president Hassan Rouhani's suggestion that the US is "playing with the lion's tail" by provoking Iran, telling his counterpart to "never, ever threaten the United States again".

Mr Rouhani had told a meeting of his country's diplomats that "America should know that peace with Iran is the mother of all peace, and war with Iran is the mother of all wars".

In a late-night response in all-caps on Twitter, the US president said Mr Rouhani would "suffer consequences the likes of which few throughout history have ever suffered before".

"We are no longer a country that will stand for your demented words of violence and death," he added. "Be cautious!"

The full comments Mr Rouhani made to his envoys, reported via the state news agency, made clear that he was talking up the prospect of peace between America and Iran, saying the White House could still change its mind after pulling out of the international nuclear deal with Tehran two months ago.

Mr Rouhani told Mr Trump that if the US moved away from efforts to build relations with Iran "this would only lead to regret". “You are not in a position to incite the Iranian nation against Iran’s security and interests,” he said.

And on Monday morning, Iran's state-owned news agency dismissed Mr Trump's warning tweet, describing it as a "passive reaction" to Mr Rouhani's remarks.

The IRNA news agency, a government mouthpiece, also described Mr Trump's message as no more than a copy of comments made by the Iranian foreign minister, Mohammad Javad Zarif, who had in the past warned the West to "never threaten an Iranian".

Iran faces increased US pressure and looming sanctions after Mr Trump’s decision to withdraw the United States from a 2015 international deal over Iran’s nuclear programme. While Mr Trump had campaigned for the White House vowing to pull the US from the deal, in the months and weeks before he made the decision, diplomats from Europe and Britain urged him to stay with it.

French President Emmanuel Macron travelled to Washington to speak with Mr Trump, saying that while the deal might not be perfect, it was the best way of ensuring Iran did not develop a nuclear weapon. Before leaving Washington during that April visit, Mr Macron had predicted that the US president would likely pull out of the deal “for domestic reasons”.

In Washington, US officials told Reuters that the Trump administration has launched an offensive of speeches and online communications meant to foment unrest and help pressure Iran to end its nuclear programme and its support of militant groups.

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Mr Rouhani and Mr Trump have also previously clashed over other subjects, including the US decision to move the US embassy to Jerusalem and a ban on citizens from certain Muslim-majority countries from travelling to the US.

But Mr Trump has not engaged in such a fierce and strongly-worded public row with another world leader, since a string of aggressive tweets against the North Korean leader Kim Jong-un ultimately presaged an historic summit between the two.