President Trump has warned the US is "targeting" 52 Iranian sites and will strike "very fast and very hard" if Tehran hits Americans or US assets.
His comments followed the US assassination of Qasem Soleimani, a top Iranian general, in a drone strike.
Iran has vowed to avenge his killing.
Mr Trump wrote on Twitter that Iran "is talking very boldly about targeting certain USA assets" in response to the general's death.
He said the US has identified 52 Iranian sites, some "at a very high level & important to Iran & the Iranian culture, and those targets, and Iran itself, WILL BE HIT VERY FAST AND VERY HARD" if Tehran strikes the US.
"The USA wants no more threats!" Trump added.
The president said the 52 targets represented the 52 Americans held hostage in Iran for more than a year from late 1979, after they were seized at the US embassy in Tehran.
Mr Trump's tweets came hours after a huge funeral procession for Soleimani was held in Baghdad, where he was killed on Friday.
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Several rocket attacks shook the area shortly after the procession, including one in the Green Zone near the US embassy.
The Iraqi military said nobody had been hurt. No group has yet said it was behind the development. Pro-Iranian militants have been blamed for other recent attacks.
With Iran already threatening harsh reprisals for the killing of the Quds Force commander, President Trump has clearly determined that the best way to de-escalate is to raise the stakes in advance, making clear what will happen if Tehran follows through on its threats.
The Trump tweet is curious in many ways – not least the symbolic mention of 52 Iranian targets being held at risk – a reference to the 52 US hostages seized in the US Embassy in Tehran back in November 1979.
His mention of targets important "to the Iranian culture" suggests a much wider target list than just leadership, military or economic sites.
President Trump is struggling to establish some kind of deterrence. But the ball is now very clearly in Iran's court and it is very hard to see how Tehran can fail to act.
Mr Trump has pursued a contradictory policy ever since he abandoned the nuclear agreement with Tehran – increasing economic pressure, threatening military action, but actually doing very little, even when Iran shot down a sophisticated US drone and struck oil installations in Saudi Arabia.
Above all, he has repeatedly underscored his and Washington's fatigue with its military involvement in the region. This as much as anything else has undermined US deterrence, something Mr Trump is now seeking, belatedly, to restore.
Why did the US kill Soleimani?
General Soleimani was widely seen as the second most powerful figure in Iran, behind Supreme Leader Ayatollah Ali Khamenei.
The 62-year-old spearheaded Iran's Middle East operations as head of the elite Quds Force, and was hailed as a heroic national figure.
But the US branded the commander and the Quds Force terroriRead More – Source
