"We are not looking for regime change. We are not looking for that at all," Trump said during a cabinet meeting. "They can't have a nuclear weapon."
The United States quit an international deal aimed at curbing Iran's nuclear program last year, hitting Tehran with crippling sanctions.
Iran said last week that it had enriched uranium past the 3.67 percent limit set by the 2015 nuclear deal, and has also surpassed the 300-kilogram cap on enriched uranium reserves.
Tensions have soared since Trump pulled out of the deal, with the US calling off air strikes against Iran at the last minute after Tehran downed an American drone, and Washington blaming the Islamic republic for a series of attacks on tanker ships.
Iranian Foreign Minister Mohammad Javad Zarif warned on Monday that the United States is "playing with fire."
According to leaked diplomatic cables published at the weekend, Britain's ambassador to Washington believed Trump pulled out of the nuclear deal because it was associated with his predecessor Barack Obama.
"The administration is set upon an act of diplomatic vandalism, seemingly for ideological and personality reasons -- it was Obama's deal," wrote the envoy, Kim Darroch, who resigned amid a storm triggered by the release of the sensitive documents.