Efforts to end the Middle East war appeared at a standstill on Tuesday, with the United States considering Tehran's latest offer to unblock the Strait of Hormuz, and Iran saying Washington could no longer dictate terms.
Iran has blockaded the waterway -- a vital conduit for oil and gas shipments -- since the start of the US-Israeli offensive two months ago, sending shockwaves through the global economy.
US officials did not dispute accounts by CNN and The Wall Street Journal that Trump was sceptical of the proposal.
Oil prices soared above $110 a barrel -- their highest level since the US-Iran ceasefire -- as the market felt new jitters over renewed conflict.
Qatar -- a US ally that was hit by Iranian strikes despite its role as a mediator -- warned of the possibility of a "frozen conflict" if a definitive resolution is not found.
"We do not want to see a return to hostilities in the region anytime soon. We do not want to see a frozen conflict that ends up being thawed every time there is a political reason," Qatari foreign ministry spokesman Majed al-Ansari said at a press conference.
The Iranian proposal, passed along by Pakistan and studied by Trump administration officials in a meeting Monday, laid out red lines including on nuclear issues and Hormuz, according to Iran's Fars news agency.
The plan would reportedly see Tehran ease its chokehold on the strait and Washington lift its retaliatory blockade on Iranian ports while broader negotiations continue, including over the thorny question of Iran's nuclear program.
Secretary of State Marco Rubio said that the proposal was "better than what we thought they were going to submit" but questioned if the Iranian officials behind it had authority, following Israeli killings of senior officials.
Rubio, in a Fox News interview, said that US demands to reopen the strait meant "going back to the way it should be" and was before the US-Israeli attack.
"They're very good negotiators," Rubio said, adding any eventual deal had to be "one that definitively prevents them from sprinting towards a nuclear weapon."
Iranian defence ministry spokesman Reza Talaei-Nik said that Washington "must abandon its illegal and irrational demands."
"The United States is no longer in a position to dictate its policy to independent nations," he said, according to state TV.
US officials including Vice President JD Vance twice last week prepared to fly to Pakistan for new talks but then stayed home.
- Trump peeved by Germany -
Trump faces domestic pressure to find an off-ramp as prices rise, with midterm elections due in November and polls showing the war is unpopular among Americans.
German Chancellor Friedrich Merz, who earlier had offered guarded support to Trump, said Monday that "the Americans obviously have no strategy" in Iran and that the war was "at the very least ill-considered".
Trump denounced Merz on social media on Tuesday, saying he "doesn't know what he's talking about".
An Iranian army spokesman told state TV on Tuesday that "we do not consider the war to be over," saying Tehran had "no trust in America."
"We have many cards that we have not yet used... new tools and methods of fighting based on the experiences of the past two wars, which will definitely allow us to respond to the enemy more decisively" should the fighting resume, Amir Akraminia said in an interview.
On a visit to Russia, Iranian Foreign Minister Abbas Araghchi said the war had shown "Iran's true power" and stability, but back home in Tehran, the mood was sober.
"Everything in the country is up in the air right now. I have not worked for a long time," small business owner Farshad told Paris-based AFP journalists.
"The country is in complete economic collapse."
- Lebanese troops said targeted -
Violence has continued on the war's Lebanese front, despite a recently extended ceasefire between Israel and Iran-backed Hezbollah, which drew Lebanon into the Middle East war by firing rockets at Israel.
Israel responded with strikes and a ground invasion.
For the first time since the ceasefire began, the Lebanese army said Tuesday that an Israeli strike had targeted its troops, wounding two soldiers in the south.
Israel's military had earlier warned residents of more than a dozen villages and towns to evacuate immediately, saying Hezbollah's "violation of the ceasefire" was compelling it to act.
The military also announced it had found and destroyed a large Hezbollah tunnel network used by elite fighters in southern Lebanon.
Despite its occupation of a swath of territory along the border, Israeli Foreign Minister Gideon Saar said his country "has no territorial ambitions in Lebanon."
Once Hezbollah and its allies "are dismantled, Israel will have no need to maintain its presence in these areas," he added.
The day before, Hezbollah's leader Naim Qassem had vowed that the armed group would "not back down."
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