Six weeks since Trump paused Operation Epic Fury for a ceasefire, talks to end the war have largely stalled.
Iran submitted a new offer to the United States this week, but its public accounts of it repeat terms previously rejected by Trump, including demands for control of the Strait of Hormuz, compensation for war damage, lifting of sanctions, release of frozen assets and the withdrawal of US troops from the area.
Trump said on Monday, and again on Tuesday, that he had come close to ordering a new bombing campaign but had put it off at the last minute to give more time for diplomacy.
"I was an hour away from making the decision to go today," Trump told reporters at the White House on Tuesday.
Iran has repeatedly threatened to retaliate for any new attacks by striking countries in the Middle East that house US bases.
On Wednesday, it suggested it would also hit targets further afield.
"If aggression against Iran is repeated, the promised regional war will extend beyond the region this time," the Revolutionary Guards said in a statement carried on state media.
Iran has largely shut the Strait of Hormuz to all ships apart from its own since the US-Israeli campaign began in February, causing the biggest disruption to global energy supplies in history.
The United States responded in April with its own blockade of Iran's ports.
Two giant Chinese tankers laden with about four million barrels of oil exited the strait on Wednesday, the latest signal that Iran is willing to ease its blockade for countries it considers friendly.
Iran had announced, while Trump was in Beijing for a summit, that it had reached an agreement to ease rules for Chinese ships.
South Korea's foreign minister said on Wednesday a Korean tanker was crossing the strait in co-operation with Iran.
Shipping monitor Lloyd's List said at least 54 ships had transited the strait on May 10-17, about double the number from the previous week, but still only a tiny fraction of the 140 or so each day that typically crossed before the war.
Trump is under pressure to end the war, with soaring energy prices hurting his Republican Party before congressional elections in November.
Since the ceasefire in late April, his public comments have veered from threats to restart bombing to declarations that a peace deal was at hand, often in the same breath.
On Tuesday he said the war would be over "very quickly".
Vice-President JD Vance, who led the US delegation in April at the only round of peace talks so far, also talked up progress: "We're in a pretty good spot here," Vance told a White House media briefing.
The US-Israeli bombing killed thousands of people in Iran before it was suspended in a ceasefire in early April.
Israel has also killed thousands more and driven hundreds of thousands from their homes in Lebanon, which it invaded in pursuit of the Iran-backed Hezbollah militia.
Iranian strikes on Israel and neighbouring Gulf states have killed dozens of people.
The Iran ceasefire has mostly held, though there was a spike in attacks on shipping and on Gulf states in early May when Trump announced a naval mission to reopen the strait.
Trump called off that mission, Project Freedom, after just 48 hours.
A new volley of drones were launched in recent days at Saudi Arabia and the UAE, which said they came from Iraq where militia allied to Iran operate.
