Tweeting in almost all capital letters late at night in the United States, Mr Trump respondent to President Rouhani's threats, saying the USA is "no longer a country that will stand for your demented words of violence and death".
Speaking to a California audience composed mostly of Iranian-Americans, Pompeo said that while the Iranian people must determine the country's direction, Washington "will support the long-ignored voice of the Iranian people".
Rouhani immediately dismissed those United States threats and on Sunday said: "You can not provoke the Iranian people against their own security and interests".
The first round of U.S. sanctions on Iran will come into force on August 6.
"Democracy will come to Iran when the people of Iran have their reckoning with their ruthless and incompetent rulers". He had also warned the USA leader not to "play with the lion's tail".
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The 2015 agreement, between Iran and the USA and other powers, had agreed to ease sanctions on Iran if the country reined in its nuclear program.
"The current U.S. president does not talk about war, however, there are credible information based on which he has attempted to force the U.S. Army to carry out military attack on Iran during the past year", Baqeri told a meeting.
Publicly, the Trump administration says its policy with Iran is not "regime change", but to change Tehran's behaviour so it stops nuclear and missile work, support for proxies in the Middle East and backing of militant groups.
"This strategy is not all sticks", said the official, who spoke on the condition that his or her name not be used. "The US is not interested in regime change and it is not really interested in improving human rights for Iranians", he added.
In an article for the New York Review of Books on Friday, NIAC president Trita Parsi echoed his colleague's assessment of Pompeo's planned speech, calling it "part of a deliberate policy of escalating tensions with Iran, targeting its economy, and supporting Iranian opposition groups-all for the goal of pressuring and destabilizing Iran".
Trump withdrew in May from the 2015 nuclear accord created to stop Iran from developing nuclear weapons.