Speaking to reporters aboard Air Force One on Saturday, Trump said the air campaign could make negotiations a moot point if all potential leaders of Iran were killed and the Iranian military was destroyed.
"At some point, I don't think there will be anybody left maybe to say 'we surrender'," Trump said.
Huge explosions were heard in parts of Tehran on Sunday, state media reported, while Israel said it had struck Iranian missile sites and command centres as the US-Israeli war against Iran entered a second week.
Iran's president apologised to neighbouring states for its attacks on US facilities in those countries in an attempt to cool anger across the Gulf, but stirred criticism from hardliners at home.
"I personally apologise to neighbouring countries that were affected by Iran's actions," Iranian President Masoud Pezeshkian said, urging them not to join US-Israeli attacks on Iran.
He dismissed Trump's demand for the Islamic Republic's unconditional surrender as "a dream", but said its temporary leadership council had agreed to suspend attacks on nearby states unless strikes on Iran originated from their territory.
Pezeshkian's comments caused a political stir in Iran, prompting his office to reiterate Iran's military would respond firmly to attacks from US bases.
Ali Larijani, Iran's secretary of the Supreme National Security Council, said on state television there was no rift among Iranian officials over the country's handling of the war.
Saudi Arabia has told Tehran continued Iranian attacks on the kingdom and its energy sector could push Riyadh to respond in kind, four people familiar with the matter told Reuters.
The governments of Saudi Arabia, Kuwait and the UAE reported Iranian drone attacks in their countries on Saturday and early Sunday with varying degrees of damage but no reported deaths.
Iran's Revolutionary Guards also targeted US forces at a base in Bahrain, Iranian state media said.
The US-Israeli attacks have killed at least 1332 Iranian civilians and wounded thousands, according to Iran's United Nations ambassador, Amir Saeid Iravani.
US forces were likely responsible for an apparent strike on an Iranian girls' school that killed scores of children, US officials have told Reuters.
Trump, without citing evidence, said Iran was responsible.
"We think it was done by Iran because they are very inaccurate as you know with their munitions," he told reporters on Saturday.
"They have no accuracy whatsoever. It was done by Iran."
Defense Secretary Pete Hegseth, standing behind Trump aboard Air Force One, said the matter was still under investigation.
Iranian attacks have killed 10 people in Israel.
At least six US service members have been killed. Their remains arrived on Saturday at an Air Force base in Delaware.
An Iranian Oil Ministry source told local media agencies fuel depots in three areas including Karaj, west of Tehran, were hit by strikes.
With the conflict spreading, Israel has launched fresh attacks in Lebanon after the Iran-aligned militia Hezbollah fired across the border.
Israel warned Lebanon of a "very heavy price" if it did not rein in Iran-allied Hezbollah militants, as it pounded the group's strongholds with air strikes and mounted a deadly airborne raid in the east.
More buildings in Beirut's Hezbollah-controlled southern suburbs were reduced to rubble, dust and tangled wires on Saturday, Reuters video showed.
The death toll from Israel's attacks on Lebanon since Monday rose to about 300 after at least four people were killed when an Israeli strike hit an apartment in the Ramada hotel building in central Beirut, Lebanon's health ministry said.
It was the first strike to hit the heart of the capital since Israel-Hezbollah hostilities resumed.
Iran's apparent strategy of maximum chaos has driven up the costs of the conflict by raising energy prices and hurting global business and logistics links.
