The Pentagon on Monday distanced itself from US President Donald Trump's assertions that he would bomb Iranian cultural sites despite international prohibitions on such attacks.

US Defence Secretary Mark Esper said the US will follow the laws of armed conflict. When asked if that ruled out targeting cultural sites, Esper said pointedly, "That's the laws of armed conflict".

The split between the president and his Pentagon chief came amid heightened tensions with Tehran following a US drone strike that killed Gen Qassem Soleimani, the head of Iran's elite Quds Force.

Trump had twice warned that he would hit Iranian cultural sites if Tehran retaliates against the US.

Esper's public comments reflected the private concerns of other defence and military officials, who cited legal prohibitions on attacks on civilian, cultural and religious sites, except under certain, threatening circumstances.

Trump first raised the prospect of targeting cultural sites in a tweet on Saturday and reiterated that view to reporters the next day.

"We have targeted 52 Iranian sites (representing the 52 American hostages taken by Iran many years ago), some at a very high level and important to Iran and Iranian culture, and those targets, and Iran itself, WILL BE HIT VERY FAST AND VERY HARD,” he tweeted.

His Twitter message caught administration officials off guard and prompted an immediate outcry from legal scholars, national security experts and Democratic lawmakers. But the president stood by his threat the following day.

"They're allowed to kill our people. They're allowed to torture and maim our people. They're allowed to use roadside bombs and blow up our people," he told reporters traveling with him on Air Force One. "And we're not allowed to touch their cultural sites? It doesn't work that way."

By international law, however, it does.

Specifically, the 1954 Hague Convention says nations must take all possible steps to protect cultural property and shall refrain from any act of hostility, directed against such property.

It also says nations must not use cultural sites for any threatening purposes that would make such locations a military target.

The Pentagon has long had a list of potential targets both inside Iran as well as those associated with Iran throughout the Middle East. Those targets and war plans are routinely updated, including during the recent uptick in hostilities.

Officials wont discuss the list, but it is certain to include an array of Iranian military sites and capabilities, including missile, air defence and command and control locations.

Any targets would go through a lengthy vetting process within the military and the Pentagon to determine that they are legal, appropriate and proportionate to any Iranian action. Only after that process is complete would a list of potential sites go before the president for approval.

Outside the Pentagon, Trump's threats were met with condemnation.

"It shows that he is somewhat deranged about this,” said US Senator Tim Kaine. "The pledge to attack cultural sites, likely, is a violation of international law."

Kaine said that all Trump is doing is escalating tensions, adding that "he seems to believe, 'I can strike you, but you can't strike me.' That's not the way the world works."

He added that Trump needs to confer with Congress.

The threats also drew reaction from the Metropolitan Museum of Art in New York.

"The targeting of sites of global cultural heritage is abhorrent to the collective values of our society," museum leaders said in a statement.

“At this challenging time, we must remind ourselves of the global importance of protecting cultural sites the objects and places by which individuals, communities, and nations connect to their history and heritage."

Ahead of Esper's comments, other administration officials made similar efforts to distance themselves from Trump without directly contradicting him.

US State Secretary Mike Pompeo said on Sunday that any US military strikes inside Iran would be legal.

"We'll behave inside the system," Pompeo said. "We always have and we always will."