While weekend peace talks in Pakistan ended with no deal, investors took heart from the fact that the two sides found some areas of agreement, with Iran saying they had been "inches away" at one point.
Soon after, the US president said the military would blockade the strategic strait -- through which a fifth of global oil and gases passes -- adding to fears for energy supplies from the Middle East.
The US military clarified it would begin blockading all Iranian ports in the Gulf from Monday at 1400 GMT, but would allow ships not coming from or heading to Iran to pass through the strait.
Trump's blockade announcement saw crude surge as much as eight percent and Asian stocks sink Monday. But optimism was revived in New York, with some observers pointing to Trump's announcement that Iranian representatives had called seeking to make a deal.
"They'd like to make a deal. Very badly, very badly," he told reporters outside the Oval Office.
While he did not identify which officials had called, all three main indexes in New York ended well in positive territory, while oil gave back all its gains.
And the optimistic tone continued into Asia, with Tokyo and Seoul leading gains thanks to renewed buying of tech firms as traders returned to the AI theme.
Taipei's 2.4 percent rise helped push the TAIEX to a fresh record high.
Hong Kong, Shanghai, Sydney, Singapore and Jakarta also joined the push higher.
London, Frankfurt and Paris rose.
Both main oil prices also fell to hold below $100, with West Texas Intermediate down more than two percent.
The equity advances extended a rebound enjoyed last week after Washington and Tehran announced a two-week ceasefire in the war, which has sent shockwaves through the world economy since it started on February 28.
SPI Asset Management's Stephen Innes said the rallies have been built on "the belief that diplomacy, however imperfect, remains in play. Last week's advance was not driven by resolution but by hope".
"The talks in Pakistan did not deliver a deal, but they did something just as important. They kept the door open. And in markets, an open door is often enough," he added.
"Oil felt that shift immediately. Not because the physical reality changed, but because the narrative did. The market began to price not the blockade itself, but the possibility that it is being used as leverage rather than a prelude to something more destructive."
Still, in a social media post, Trump said the bulk of Iran's navy had already been destroyed but that if any of what he said were Tehran's few remaining "fast attack ships" approach the blockade "they will be immediately ELIMINATED".
He also said 34 ships had passed through the strait Sunday, adding it was the most since the war began, though the figure could not be immediately corroborated.
Iran continued to hit out at Washington, with Foreign Minister Abbas Araghchi blaming it for the impasse in the talks during a call with his Saudi counterpart Prince Faisal bin Farhan.
"Unfortunately, we witnessed the continued excessive demands of the American side in the negotiations, which led to the failure to achieve a result," his ministry quoted him as saying.
Analysts suggested the US president was trying to starve Iran of funds but also pressure Beijing, the biggest buyer of Iranian oil, to lean on Tehran to reopen Hormuz.
Meanwhile, the head of the International Energy Agency warned Monday that April could prove a tougher month than March for energy markets and the economy.
Fatih Birol said March saw delivery of cargo loaded before the crisis in the Middle East, but "during the month of April, nothing has been loaded".
"The longer the disruption is, the more severe the problem becomes," he told reporters.
- Key figures at 0715 GMT -
West Texas Intermediate: DOWN 2.7 percent at $96.45 a barrel
Brent North Sea Crude: DOWN 1.1 percent at $98.26 a barrel
Tokyo - Nikkei 225: UP 2.4 percent at 57,877.39 (close)
Hong Kong - Hang Seng Index: UP 1.0 percent at 25,908.56
Shanghai - Composite: UP 1.0 percent at 4,026.63 (close)
London - FTSE 100: UP 0.2 percent at 10,605,08
Euro/dollar: UP at $1.1772 from $1.1761 on Monday
Pound/dollar: UP at $1.3524 from $1.3507
Dollar/yen: DOWN at 159.19 yen from 159.41 yen
Euro/pound: DOWN at 87.05 pence from 87.08 pence
New York - Dow Jones: UP 0.6 percent at 48,218.25 (close)
dan/mtp