Trump said in a telephone interview with Reuters it was very early in the process of picking a new leader but that the late Supreme Leader Ayatollah Ali Khamenei's son Mojtaba was an unlikely choice, without elaborating.
"We want to be involved in the process of choosing the person who is going to lead Iran into the future," Trump said.
"We don't have to go back every five years and do this again and again ... Somebody that's going to be great for the people, great for the country."
Trump also encouraged Iranian Kurdish forces to go on the offensive.
"I think it's wonderful that they want to do that, I'd be all for it," the president said.
When asked if the US would provide or had offered air cover, he responded, "I can't tell you that" but added that the objective for the Kurds would be "to win".
"If they're going to do that, that's good," Trump added.
Iranian Kurdish militias have consulted with the United States in recent days about whether, and how, to attack Iran's security forces in the western part of the country, according to three sources with knowledge of the matter.
The Iranian Kurdish coalition of groups based on the Iran-Iraq border in the semi-autonomous region of Iraqi Kurdistan has been training to mount such an attack in hopes of weakening the country's military, as the United States and Israel pound Iranian targets with bombs and missiles.
Trump also signalled confidence that the major shipping route near Iran, the Strait of Hormuz, will remain open.
Closing the Strait of Hormuz, a narrow choke point between Iran and Oman through which a fifth of the world's crude oil and liquefied natural gas passes, has been one of Iran's main objectives, and shipping through the crucial energy artery has ground to a near halt after Iranian hits on six vessels.
"They have no navy, you know the navy is now at the bottom of the sea," Trump said.
"I'm watching Hormuz very closely."
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