Israeli Defence Minister Israel Katz said meanwhile that Iran's Supreme Leader Ayatollah Ali Khamenei could face the same fate as Iraqi President Saddam Hussein, who was toppled in a US-led invasion and eventually hanged after a trial.
"I warn the Iranian dictator against continuing to commit war crimes and fire missiles at Israeli citizens," Katz told top Israeli military officials.
Explosions were reported in Tehran and in the city of Isfahan in central Iran, while Israel said Iran had fired more missiles towards it and air raid sirens sounded in Tel Aviv and southern Israel.
The Israeli military said it had conducted strikes on 12 missile launch sites and storage facilities in Tehran.
Trump predicted on Monday that Israel would not be easing its attacks on Iran launched on Friday.
But he also said he might send US Middle East Envoy Steve Witkoff or Vice President JD Vance to meet Iranian officials.
Trump had said his early departure from a G7 summit in Canada had "nothing to do with" working on a ceasefire between Israel and Iran, and that something "much bigger" than that was expected.
In a further post on Tuesday, the US president said Khamenei's whereabouts were known but "we are not going to take him out (kill!), at least not for now," while adding: "Our patience is wearing thin."
Vance said the decision on whether to take further action to end Iran's uranium enrichment program, which some countries suspect is aimed at developing a nuclear bomb, "ultimately belongs to the president".
UK Prime Minister Keir Starmer said there was no indication that the US was about to enter the conflict.
Trump had not yet decided whether the US military will intervene on Israel's side, German Chancellor Friedrich Merz said.
Trump met for 90 minutes with his National Security Council on Tuesday afternoon to discuss the conflict, a White House official said.
Details were not immediately available.
Khamenei's main military and security advisers have been killed by Israeli strikes, leaving major holes in his inner circle and raising the risk of strategic errors, according to five people familiar with his decision-making process.
The Israeli military said Iran's military leadership was "on the run" and that it had killed Iran's wartime chief of staff Ali Shadmani overnight, four days after he had replaced another top commander killed in the strikes.
With Iranian leaders suffering their most dangerous security breach since the 1979 Islamic Revolution that toppled a US-backed monarch and led to clerical rule, the country's cyber security command banned officials from using communications devices and mobile phones, Fars news agency reported.
Israel had launched a "massive cyber war" against Iran's digital infrastructure, Iranian media reported.
Israel launched its air war after saying it had concluded that Iran was on the verge of developing a nuclear weapon.
Iran denies seeking nuclear weapons and has pointed to its right to nuclear technology for peaceful purposes, including enrichment, as a party to the Non-Proliferation Treaty.
Israel, which is not a party to the NPT, is the only country in the Middle East widely believed to have nuclear weapons.
Israel does not deny or confirm that.
Israeli Prime Minister Benjamin Netanyahu has stressed that he will not back down until Iran's nuclear development is disabled, while Trump says the Israeli assault could end if Iran agrees to strict curbs on enrichment.
Israel says it now has control of Iranian air space and intends to escalate the campaign in the coming days.
Iran has so far fired nearly 400 ballistic missiles and hundreds of drones towards Israel, with about 35 missiles penetrating Israel's defensive shield and making impact, Israeli officials say.
Iran's Revolutionary Guards said they had hit Israel's Military Intelligence Directorate and foreign intelligence service Mossad's operational centre early on Tuesday.
There was no Israeli confirmation of such attacks.
Iranian officials have reported 224 deaths, mostly civilians, while Israel said 24 civilians had been killed.
with DPA
