Investors were also gearing up for key central bank meetings and earnings reports from Wall Street giants this week.
Tehran was reported to have passed “written messages” to Washington via Pakistan spelling out its red lines in peace talks, including on its nuclear programme and the future of the crucial waterway.
The White House said the US president and his met Monday to discuss the offer but spokeswoman Karoline Leavitt refused to say if Trump would accept the proposal.
Iran’s proposed interim deal is said to see it reopen the Strait of Hormuz – through which a fifth of oil and LNG usually flows – in exchange for Washington ending its blockade of Iranian ports.
The plan also postpones more complex negotiations over its nuclear programme, a major sticking point for Trump.
Hopes for a deal had been rising going into last weekend but Trump dashed them on Saturday by scrapping a planned trip by his envoys Steve Witkoff and Jared Kushner to Islamabad.
Iran’s envoy to the United Nations Amir Saeid Iravani told a Security Council session the country would first need guarantees Washington and Israel would not attack again if it was to offer security assurances in the Gulf.
But secretary of state Marco Rubio said Iran’s stance on the Strait of Hormuz did not meet US demands.
“If what they mean by opening the straits is, ‘yes, the straits are open as long as you coordinate with Iran, get our permission or we’ll blow you up and you pay us,’ that’s not opening the straits,” Rubio told Fox News.
Meanwhile, president Vladimir Putin told Iran’s foreign minister Abbas Araghchi that Russia would do everything it could to halt the Middle East war, as the two met in Saint Petersburg.
Oil prices extended gains Tuesday, with Brent heading back towards US$110 a barrel, while stock markets struggled.
Tokyo, Hong Kong, Shanghai and Sydney led losses, though there were gains in Seoul, Singapore, Taipei and Jakarta.
That came after the S&P 500 and Nasdaq once again hit record highs in New York.
IG analyst Tony Sycamore pointed out that Tehran could be more willing to strike a deal soon as its ageing storage facilities were expected to hit maximum capacity this week.
He added that “if forced shut‑ins follow, Tehran risks irreversible long‑term damage to its reservoirs and a serious hit to future production and revenue streams”.
However he said that while Iran’s latest offer was a positive, “it is hard to see the US accepting anything less than a comprehensive deal that both opens the Strait of Hormuz and addresses Iran’s nuclear weapons programme”.
Focus is also on a host of central bank meetings this week, with the Bank of Japan seen holding pat on interest rates later Tuesday.
The Federal Reserve, European Central Bank and Bank of England are expected to follow suit amid growing concerns about a fresh spike in inflation caused by the surge in energy costs.
Tech giants Apple Meta Platforms and Microsoft are also due to report, as are older industrial companies including Ford and ExxonMobil.