Negotiations between the US and Iran to reach a final deal to end the war are expected to continue throughout the night in Switzerland, a senior US diplomat engaged in the talks said.
The US-Iran talks began on Sunday, after last week's agreement to reach a deal in 60 days.
Late Sunday night, the US diplomat said discussions were focused on "clarifying some of the confusing messaging from Iran" on the re-opening of the Strait of Hormuz, enforcing the ceasefire in southern Lebanon and "elements" of a nuclear deal.
Earlier, Trump had threatened to attack Iran if it did not stop Hezbollah amid continued clashes with Israel in Lebanon. Iran dismissed the warning, saying it was prepared to fight.
The US diplomat said both delegations, who are meeting in the Swiss city Lucerne, would use "today's work as a starting point for ongoing technical talks going forward".
Last week's initial agreement includes a commitment to reach a final deal within 60 days, as well as an end to fighting on "all fronts" - including in Lebanon - and the reopening of the Strait of Hormuz.
However, there has since been an upsurge in fighting between Hezbollah and Israeli forces in southern Lebanon, and Israeli air strikes that the health ministry says have killed dozens of Lebanese including women and children.
That escalation led the US to declare a new ceasefire between Israel and Hezbollah on Friday. Continued clashes and air strikes prompted Iran on Saturday to announce it had shut the Strait of Hormuz - though tracking data shows vessels have continued to pass through it.
As the talks in Lucerne between the US and Iranian delegations began, Trump posted that Iran "must immediately stop their highly paid PROXIES in Lebanon from causing trouble" and threatened to "hit Iran very hard again" if they did not.
Iran's lead negotiator Mohammad Bagher Ghalibaf responded by saying: "Don't they think that if their threats had any effect, they wouldn't be in this desperate situation today?... No matter how much they talk, it is we who take action."
On Sunday, fighting was reported to have diminished but Israeli Prime Minister Benjamin Netanyahu insisted that the Israeli military would remain in southern Lebanon for as long as is necessary to protect northern Israel.
Hezbollah leader Naim Qassem has rejected any Israeli military presence in southern Lebanon and said Hezbollah would defend itself.
Speaking before the talks at the Swiss resort of Bürgenstock, US lead negotiator Vice-President JD Vance said Trump had asked negotiators to "turn over a new leaf".
He added that if Iran's leadership was willing to give up being a "driver of regional instability" and its "nuclear weapons ambitions for the longer term", then the US "is willing to fundamentally transform our relationship with that country".
Iran has insisted its nuclear programme is peaceful.
Vance was joined by Trump's son-in-law Jared Kushner and special envoy Steve Witkoff. Ghaibaf was accompanied by Iran's Foreign Minister Abbas Araghchi.
Also in Switzerland were Pakistan's prime minister and army chief and the Qatari prime minister. Pakistan has acted as a mediator throughout the war and hosted a previous round of negotiations between the US and Iran.
Qatar has also mediated and late on Sunday Qatari Prime Minister Sheikh Mohammed bin Abdul Rahman Al Thani said he welcomed the continuation of US-Iran talks.
The US and Iranian presidents signed the initial agreement earlier this week, aiming to end the war with immediate effect.
Under the deal, Iran was to reopen the Strait of Hormuz, the key shipping channel through which 20% of the world's oil and natural gas travels. Its effective closure had driven up fuel prices and disrupted the global economy.
The US also agreed to lift a military blockade on ships going to and from Iranian ports.
The initial deal also included a $300bn (£224bn) plan for Iran's "reconstruction", and the US terminating "all types of sanctions" on it.
But the issue of Iran's nuclear programme, the main reason stated by the US for the conflict, is still to be negotiated. In Trump's first term, he withdrew the US from an Obama-era nuclear deal with Iran and reimposed economic sanctions.
On Sunday some vessels appeared to be entering, exiting and transiting the strait, according to location data on the maritime tracking website MarineTraffic, despite Iran's claim - disputed by the US - to have closed the strait.
Four tankers passed through the strait by late afternoon. Four other vessels, all cargo or carrier ships, appear to be heading eastwards after being stationary on the western approach to the strait since the end of last week. In the other direction, four vessels appear to be heading out of the strait westwards having departed waters off the Iran coast.
Tracking data may not capture all movements as some vessels may have turned their trackers off.
The initial deal also called for fighting to stop on all fronts, but in Lebanon Israeli air strikes have since killed at least 67 people, while Hezbollah attacks have killed five Israeli soldiers.
Israel has insisted that its conflict with Hezbollah is separate from the war on Iran, which it mounted alongside the US on 28 February.
Lebanon was drawn into the war shortly afterwards, when Hezbollah launched rockets into Israel in retaliation for a strike that killed Iran's supreme leader.
Israel responded by launching a bombing campaign across Lebanon and occupying around 5% of the country's territory in the south - hoping to drive back Hezbollah fighters from its northern border - and has said it has no intention of withdrawing.
Since 2 March, 4,057 people have been killed in Lebanon, the country's health ministry says. At least 34 Israeli soldiers have been killed in Lebanon, and four civilians in northern Israel, Israeli authorities say.
Additional reporting by Lana Lam, Emma Pengelly and Richard Irvine-Brown
