US President Donald Trump threatened Iran's President Hassan Rouhani with consequences the likes of which few throughout history have suffered before," in a tweet.
Tweeting in nearly all capital letters late at night in the United States, Mr Trump respondent to President Rouhani's threats, saying the US is "no longer a country that will stand for your demented words of violence and death".
He finished the tweet with "be cautious!"
Earlier, Iran's president warned Mr Trump against provoking his country, while indicating peace between the two nations might still be possible.
Mr Trump this year pulled the US out of the international deal meant to prevent Tehran from developing a nuclear weapon and ordered increased American sanctions.
Iran's semi-official ISNA news agency quoted Mr Rouhani as warning Mr Trump on Sunday: "Do not play with the lion's tail or else you will regret it."
Mr Trump has suggested Iranian leaders are "going to call me and say 'let's make a deal'", but Iran has rejected talks.
Mr Rouhani suggested peace was still possible and said: "America must understand well that peace with Iran is the mother of all of peace, and war with Iran is the mother of all wars."
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