The U.S. Embassy in Qatar urged Americans currently in the Middle East to exercise "heightened vigilance and be aware of potential flight cancellations, airspace closures, and travel disruptions".
The six-month-old conflict had shifted into an economic standoff before the weekend strikes, in which U.S. forces attacked Iran's Larak Island on Sunday and Iran responded by launching missiles at two U.S. air bases in Jordan.
Although he warned of a U.S. response to the Iranian strikes, Trump also told reporters on Monday that they did not signal a return to full-scale war, and a senior Iranian source told Reuters they were a "limited and contained confrontation".
But pledges by both sides to respond to further attacks have highlighted how quickly a conflict that has increasingly been fought through sanctions, blockades and economic pressure could spill back into military action.
Iranian President Masoud Pezeshkian had said early on Tuesday at a regional summit in Kyrgyzstan that Tehran would reciprocate immediately if Washington honoured its commitments under a June Memorandum of Understanding that was intended to halt the conflict.
The MOU declared an end to fighting that has now killed thousands in Iran and Lebanon since U.S. and Israeli strikes began on February 28. But it also deferred many of the most difficult issues and paved the way for a broader 60-day negotiation period — which passed without further agreement.
Iranian officials have shown no public sign since then of bowing to pressure.
"If the enemy wants us not to export oil from the Persian Gulf, no one will be able to export oil," Iranian parliament speaker Mohammad Baqer Qalibaf was quoted as saying by Iranian media.
Foreign Ministry spokesman Esmaeil Baghaei told a press conference in Tehran that Washington was "addicted to making excessive demands and has mistaken negotiations for dictating terms".
"As long as they act in this manner, they will certainly get no result other than what they have achieved so far," he added, warning that Tehran would "use all our capabilities, whether on the battlefield or through our diplomatic apparatus".
Before the latest attacks, Bessent had said Washington was likely to announce bank sanctions against Iran this week and next, stepping up an attempt to "economically asphyxiate" Iran's leadership after six months of conflict.
At a G20 finance leaders' meeting in Asheville, North Carolina, he said the U.S. was also looking at targeting airline leasing companies and other entities that did business with the IRGC.
"We have zero tolerance. We are going to economically asphyxiate this regime," he said.
Iran's leaders have defied decades of U.S. sanctions, many of which have been aimed at curtailing Tehran's oil revenues and acquisition of weapons components.
Bessent has also warned that countries doing business with Iran could face U.S. sanctions.
The heightened tensions have renewed fears of oil supply disruptions from the world's key crude-producing region.
"The tit-for-tat missile exchanges between the U.S. and Iran bring validation to those who believe that even if not a ‘forever war’, this conflict will run and run," said PVM analyst John Evans.
Highlighting such concerns, two supertankers carrying Saudi oil were struck by unknown projectiles within minutes of each other while transiting outbound through the Strait of Hormuz late on Monday, according to data from shipping intelligence and tracking firms Marisks and Kpler.
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