The US president said that "despite requests" from unnamed individuals, he had "no plans" to meet his Iranian counterpart
"I am sure he is an absolutely lovely man," Mr Trump added in a tweet on Tuesday morning.
The comments come amid a diplomatic standoff between Washington and Tehran after Mr Trump pulled out of an international nuclear deal with America's historic foe in May.
He has since begun to reinstate economic sanctions and has pressed other countries to stop buying Iranian oil, prompting criticism in Iran that the US was waging economic war.
Mr Rouhani has also ruled out a meeting, telling NBC News on Monday that conditions had not been met for talks to take place.
"Naturally, if someone is keen on having a meeting and holding dialogue and creating progress in relationships, that person would not use the tool of sanctions and threats [and bring] to bear all of its power against another government and nation," he told the US broadcaster.
Mr Trump's tweet also comes as world leaders gather at the UN in New York, where he will address the General Assembly later on Tuesday.
He is expected to be unapologetic about his decision to engage with North Korea and remove the US from the Iran nuclear accord.
Ahead of his UN address, Iran's Revolutionary Guards called Mr Trump "evil and adventurous" and said his "economic war and cruel sanctions" were an attack on their country.
The Independent
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