Amid the rapidly-spreading protests, Iran's parliament announced Wednesday it would hold a special session to question Rouhani about the plummeting currency and struggling economy.
As the Trump administration readies to re-impose sanctions on Iran that were lifted under the 2015 nuclear accord, America's European allies fear greater regional instability.
Referring to recent sporadic protests in Iranian cities, Pompeo said: "The Iranian people are not happy - not with the Americans but with their own leadership".
It's not inconceivable a meeting on the sidelines could occur then - Tehran will be looking anxiously to a November deadline for oil buyers to stop purchasing Iranian crude.
After the official statement, the US President signed the Memorandum on the resumption of sanctions against Iran.
Following the USA exit, Iran and the remaining parties launched talks to save the accord.
In Washington, U.S. Secretary of State Mike Pompeo said the White House would make an announcement on Monday detailing the reimposition of sanctions on Iran that Trump has ordered for this week. "I do believe that they will probably end up wanting to meet".
The United States has told third countries they must halt imports of Iranian oil from early November or face us financial measures.
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In response, Donald Trump sent an explosive, all-caps tweet telling Mr Rouhani to "never, ever threaten the United States again" and "be cautious!".
This past week, though, Trump said he'd be willing to meet with Iranian leaders without any preconditions.
Mr Pompeo, in a speech to a largely Iranian-American audience in California, dismissed Mr Rouhani and Foreign Minister Javad Zarif, as "polished front men for the ayatollahs' global con artistry".
If the re-imposed sanctions caused the government in Tehran to collapse, Iran would likely devolve into civil war like what unfolded in Syria or radicals would assume power, the diplomat said.
APIf Trump helped North Korea's image, imagine what he could do for Iran.
And John Bolton, the president's national security adviser, is a well-known Iran hawk who has advocated for regime change.
"Iran, and it's economy, is going very bad, and fast!" Threats, sanctions & PR stunts won't work.
In his first public comments after the Trump remarks, Iranian President Hassan Rouhani avoided mentioning Trump's comments, instead stressing the need for the other nations involved in the nuclear deal to forge ahead with their pledges of trying to salvage it.