President Trump has warned the US is "targeting" 52 Iranian sites and will strike "very fast and very hard" if Tehran attacks Americans or US assets.
The president's remarks followed the US assassination of Qasem Soleimani, a top Iranian general, in a drone strike.
Soleimani's killing was a major escalation between the two nations, and Iran vowed to take "severe revenge".
Writing on Twitter, Mr Trump accused Iran of "talking very boldly about targeting certain USA assets".
He said the US had identified 52 Iranian sites, some "at a very high level & important to Iran & the Iranian culture", and warned they would be "HIT VERY FAST AND HARD" if Tehran struck at the US.
The president said the targets represented 52 Americans who were held hostage in Iran for more than a year from late 1979 after they were taken from the US embassy in Tehran.
Shortly after the president's tweets were posted, the website of a US government agency appeared to have been hacked by a group calling itself "Iran Cyber Security Group Hackers". A message on the American Federal Depository Library Programme site read: "This is a message from the Islamic Republic of Iran.
"We will not stop supporting our friends in the region: the oppressed people of Palestine, the oppressed people of Yemen, the people and the Syrian government, the people and government of Iraq, the oppressed people of Bahrain, the true Mujahideen resistance in Lebanon and Palestine, [they] will always be supported by us."
The web page contained a doctored image of President Trump, depicting him being hit in the face and bleeding at the mouth. "This is only small part of Iran's cyber ability!" read text on the site.
What happened earlier on Saturday?
Mr Trump's tweets followed a huge funeral procession for General Soleimani held in Baghdad, where he was killed in a targeted drone strike on Friday as he left the airport in a convoy. Mourners waved Iraqi and militia flags and chanted "death to America".
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 immediately claimed responsibility for the rocket attacks. Pro-Iranian militants have been blamed for other recent attacks.
Soleimani's body arrived back in Iran on Sunday, the country's IRIB news agency reported.
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