In an interview with Reuters on Thursday, Trump said Mojtaba Khamenei, the son of the late Supreme Leader Ali Khamenei - a hardliner who has been considered a favourite to succeed his father - was an unlikely choice.
"We want to be involved in the process of choosing the person who is going to lead Iran into the future," he 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," Trump said.
He would not say whether the United States would provide air cover for any Kurdish offensive.
Two drone attacks targeted an Iranian opposition camp in Iraqi Kurdistan on Thursday, as well as an oil field operated by an American firm, security sources said.
The comments came as the Israeli military warned residents to evacuate areas including eastern Tehran, while Iranian media reported blasts were heard in various parts of the capital. An air attack killed 17 people in a guest house on a road northwest of the capital, Iranian state television said.
With the war now in its sixth day, Iran launched a series of attacks on Israel, the United Arab Emirates and Qatar. Fire crews in Bahrain extinguished a blaze at a refinery following a missile strike.
In Iran, at least 1230 people have been killed, according to the Iranian Red Crescent Society, including 175 schoolgirls and staff killed at a primary school in Minab in the country's south on the first day of the war. Another 77 have been killed in Lebanon, its Health Ministry said.
Thousands of people fled southern Beirut on Thursday after Israel warned residents to leave.
On Thursday, Iran's Revolutionary Guards said they had hit a US tanker in the northern part of the Gulf and the vessel was on fire, the latest of numerous reports of such attacks.
Visiting an air force base in the south of the country, Israeli Prime Minister Benjamin Netanyahu said Israel's achievements so far in Iran had been "great" but that "much work still lies ahead".
Iran's foreign minister said Washington would "bitterly regret" the precedent it had set by sinking a ship in international waters without warning. A commander of the Revolutionary Guards, General Kioumars Heydari, told state TV: "We have decided to fight Americans wherever they are."
The body of Supreme Leader Ayatollah Ali Khamenei, killed in the first hours of the US-Israeli air campaign in the first assassination of a country's top ruler by an airstrike, had been due to lie in state in a Tehran prayer hall from Wednesday evening to launch three days of mourning.
But the memorial, expected to draw many thousands of mourners to the streets, was abruptly postponed.
Two sources familiar with Israel's battle plans said Israel, having killed many Iranian leaders, was now planning to enter a second phase when it would target underground bunkers where Iran stores its missiles.
Israel has said its aim is to overthrow Iran's clerical rulers. Washington says its goal is to prevent Tehran from being able to project force beyond its borders, but it has also called on Iranians to rise up and seize power.
Israel had already made the decision to kill Khamenei back in November, Defence Minister Israel Katz said on Thursday.
Although the number of missiles fired at Israel has dropped since Saturday, Iranian and Hezbollah barrages still sent millions rushing to shelters as air raid sirens sounded and schools remained shut.
