The American president delivered the incendiary message via his Twitter account.
On Friday Soleimani, head of Iran’s Al-Quds Force, was killed in an airstrike near Baghdad airport.
Soleimani reportedly had connections with pro-Iranian militia and terrorist groups, like Lebanon’s Hezbollah, across the Middle East.
His assassination followed an attack on the American Embassy in Baghdad by rioters affiliated to an Iranian backed militant group.
Iran has vowed to avenge Soleimani’s killing.
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However on Twitter Trump promised a swift American response if US targets are hit by Tehran.
He wrote: “Iran is talking very boldly about targeting certain USA assets as revenge for our ridding the world of their terrorist leader who had just killed an American, badly wounded many others, not to mention all of the people he had killed over his lifetime including recently hundreds of Iranian protestors.
“He was already attacking our Embassy, and preparing for additional hits in other locations.
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“Iran has been nothing but problems for many years.
“Let this serve as a WARNING that if Iran strikes any Americans, or American assets, we have targeted 52 Iranian sites (representing the 52 American hostages taken by Iran many years ago), 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.”
Other social media users responded by pointing out deliberately targeting cultural sites is against international law.
The UK’s Defence Secretary said on Saturday that the US had the right to defend itself in relation to the Soleimani killing.
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Ben Wallace said: “We urge all parties to engage to de-escalate the situation.
“Under international law the United States is entitled to defend itself against those posing an imminent threat to their citizens.”
Mr Wallace also said two Royal Navy warships will escort UK flagged shipping through the Strait of Hormuz.