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.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.
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 terrorists, holding them responsible for the deaths of hundreds of US personnel.
Speaking on Friday afternoon, President Trump said Soleimani was “plotting imminent and sinister attacks” on US diplomats and military personnel in Iraq and elsewhere in the region.
The general was killed by an air strike at Baghdad airport early on Friday, on the orders of President Trump. The president said the action was taken to stop, not start, a war.
On Saturday the group issued a warning to Iraqi security forces to “stay clear of American bases by a distance not less [than] 1,000m (0.6 miles) starting Sunday evening”, al-Mayadeen TV reported.
In response to Iranian threats of revenge, the US has sent 3,000 more troops to the Middle East and advised its citizens to leave Iraq.