The interview, appearing on CNBC's "Closing Bell" on Monday, came two hours after Trump said during a press conference with Italy's prime minister that he would meet with Iran's leaders, without pre-conditions, "anytime they want."
The secretary of State offered a more qualified take on the potential for top-level meetings with Iran, listing several pre-conditions.
"We've said this before," Pompeo said. The "president wants to meet with folks to solve problems. If the Iranians demonstrate a commitment to make fundamental changes in how they treat their own people, reduce their malign behavior, can agree that it's worthwhile to enter into a nuclear agreement that actually prevents proliferation, then the president said he's prepared to sit down and have the conversation with them."
U.S.-Iran tensions have intensified of late, after Trump tore up an Obama-era nuclear deal with the Middle Eastern nation and as the administration prepares to fully re-impose sanctions on the regime.
It is far from certain Iran would agree to the terms outlined by Pompeo on Monday.
In his first major foreign policy speech as secretary of State two months ago, Pompeo issued a list of 12 demands that would require Iran to essentially reverse several pillars of its long-standing foreign policy. Iranian President Hassan Rouhani called the demands "not acceptable."
The Trump administration wants to tie a new nuclear agreement to limits on Iran's ballistic missile program and its role in regional conflicts in Syria, Iraq, Yemen and elsewhere. It also wants to make permanent key aspects of the original accord that expire 10 to 15 years from the start of the deal. Iran has rejected those conditions.
Earlier this month, Rouhani's chief of staff claimed that Iran rejected eight requests by the Trump administration to hold meetings between the nations' leaders on the sidelines of a UN General Assembly meeting last year.
Asked about the claim, Pompeo said he would not speak about private conversations that may or may not have taken place.
The first of two deadlines for foreign countries to wind down business ties with Iran arrives in just one week. The Trump administration is also pushing oil buyers to stopping importing supplies from Iran, the world's fifth biggest crude producer, by the second deadline at the start of November.
Iran has recently threatened to shut down the Strait of Hormuz, the world's busiest sea lane for oil exports, if the United States goes through with the oil sanctions. Iran's Houthi rebel allies in Yemen forced Saudi Arabia to suspend oil shipments through a nearby strait in the Red Sea last week after they attacked a pair of Saudi tankers.
The secretary of State spoke with CNBC's Michelle Caruso-Cabrera hours after he announced $113 million in new Trump administration investments in Asia.