US President Donald Trump has threatened Iran against any retaliation, saying US "will go after" other targets in Iran after strikes that he said "obliterated" Iranian nuclear sites.

"There will be either peace or there will be tragedy for Iran far greater than we have witnessed over the last eight days," Trump said in a nationally televised speech at the White House early on Sunday.

"A short time ago, the US military carried out massive precision strikes on the three key nuclear facilities in the Iranian regime, Fordow, Natanz, and Esfahan. Everybody heard those names for years as they built this horribly destructive enterprise," Trump said.

"Our objective was the destruction of Iran's nuclear enrichment capacity and a stop to the nuclear threat posed by the world's number one state sponsor of terror," he added.

Trump called the US strikes "a spectacular military success." He added: "Iran's key nuclear enrichment facilities have been completely and totally obliterated."

The US President called Iran "the bully of the Middle East", saying Tehran "must now make peace".

"If they do not, future attacks will be far greater and a lot easier."