Facing strong criticism that such attacks would be a war crime, Trump said he was "OK" with following international law. However, he repeated an earlier complaint that he found the restriction unfair.

"Think of it: they kill our people, they blow up our people and then we have to be very gentle with their cultural institutions. But I'm OK with it," Trump told reporters.

"You know what, if that's what the law is, I like to obey the law."

He added a warning that if Iran "does anything that they shouldn't be doing, they're going to be suffering the consequences and very strongly."

Trump previously prompted an outcry from domestic critics, the Iranian government and the UN'S cultural agency UNESCO when he said that he did not need to abide by international law on protecting such sites in war.

"It doesn't work that way," he said.

Trump's Secretary of State Mike Pompeo and Defense Secretary Mark Esper had both sought to walk back the threat.

Iran boasts an ancient culture with two dozen places on UNESCO's list of world heritage sites.