By Humeyra Pamuk, Jana Choukeir, Ahmed Tolba and Steve HollandZURICH/DUBAI/WASHINGTON, June 21 (Reuters) - Negotiators from the United States and Iran were set to start peace talks in Switzerland on Sunday, even as U.S. officials disputed Iranian claims about closing the key Strait of Hormuz.A high-level Iranian team arrived in Switzerland for peace talks with the U.S., Iranian state media reported, as U.S. Vice President JD Vance departed Washington for meetings that Pakistan said will begin on Sunday.Although the U.S. and Iran had agreed to a 60-day ceasefire while negotiations take place, Tehran's Islamic Revolutionary Guard Corps on Saturday declared the Strait of Hormuz shut, though the U.S. military said commercial vessels had continued operating in the waterway.Those developments could complicate talks in which both sides seek to advance an interim deal brokered by Pakistan and signed on Wednesday by U.S. President Donald Trump and Iranian President Masoud Pezeshkian to end their almost four-month war.Pointing to what it called Israeli "crimes" in Lebanon that violated U.S. commitments to a ceasefire, the IRGC warned ships would be at risk if they approached the Strait, a vital conduit for global oil and gas supplies.But U.S. Central Command said 55 merchant ships transited the strait on Saturday with more than 17 million barrels of oil for global markets.U.S. forces will ensure commercial traffic continues, Central Command added.Trump in a social media post on Saturday wrote that no toll will be charged for passage through the Strait during or after the 60-day ceasefire — unless the U.S. imposes one should peace talks fail.Trump left open the possibility of a Hormuz toll levied by the United States "for services rendered as the Guardian Angel to the countries of the Middle East" if a peace deal is not completed.Mohammad Mokhber, an adviser to Iranian Supreme Leader Ayatollah Mojtaba Khamenei, accused the U.S. on X of failing to implement the first clause of its 14-point interim deal with Iran, which includes a ceasefire "on all fronts," including Lebanon.He said that, as long as the agreement was only on paper, the flow of Middle East energy would remain halted.The Lebanon truce appeared fragile as Israeli forces and the Iran-backed militant group Hezbollah attacked each other.MOMENTUM BUILDS FOR U.S.-IRAN TALKS IN SWITZERLANDThe Iranian delegation was led by chief negotiator Mohammad Baqer Qalibaf and included Foreign Minister Abbas Araqchi as well as senior security, central bank and oil officials, Iranian media said. In addition to Vance, the U.S. negotiating team includes envoys Steve Witkoff and Jared Kushner, Trump's son-in-law.Iranian Foreign Ministry spokesperson Esmaeil Baghaei said Iran would press in Switzerland for fulfilment of commitments, citing past failures by the other side to honour agreements.Vance, in an interview with Fox News, said he was confident the ceasefire would hold, and that he had seen no evidence that the Strait of Hormuz was closed.The U.S. vice president left for Switzerland shortly after 4 p.m. ET (2100 GMT) on Saturday. Negotiators would likely have a “couple days of talks," Vance told reporters before boarding a plane at Joint Base Andrews in Maryland."I think we're going to hopefully make progress on the nuclear issue, make progress on the Lebanon ceasefire issue," he said.A halt to fighting in Lebanon was one of the conditions for starting U.S.-Iranian talks on Tehran's nuclear program and other issues. But Lebanese Civil Defence said that 20 people had been killed by Israeli strikes in Lebanon on Saturday, hours after a truce there took effect.Israel said it was responding to attacks from Hezbollah, while the Iran-backed group said it would not allow Israel "freedom of movement" in Lebanon.Israel, left out of the talks, has said it is not party to the Iran-U.S. deal, and will keep its forces in the Lebanese territory it occupies.The Israeli broadcaster Channel 12 reported that the prime minister and defense minister instructed the military to hold fire in Lebanon, but that it would not withdraw from areas it had captured.A poll exclusively shared with Reuters and conducted by the Israeli Hebrew University found that some 92% of Israelis believe Iran benefited more from the joint Israeli-U.S. military campaign than Israel and only some 8% of Israelis think Israel emerged victorious. Almost 90% of Israelis said the goals of the war were not met and more than 70% don't believe Netanyahu's claims that there were major achievements.Lebanon's state news agency NNA said Israeli warplanes and drones had struck locations across southern Lebanon and the Bekaa Valley on Saturday, both Hezbollah strongholds.An Israeli military official said Hezbollah fired more than 50 projectiles at Israeli forces in southern Lebanon overnight, and that Israel had attacked what it described as Hezbollah targets in response.A military statement said Israel was committed to the ceasefire but would continue to act against any threat to Israel or its forces.Lebanon's health ministry says 4,057 people have been killed in Israeli attacks since March 2, including medics, women and children, though it does not specify how many of the dead were combatants.Israeli authorities say at least 32 soldiers and four civilians have been killed in fighting with Hezbollah. (Reporting by Reuters bureaus; Writing by William Maclean and John Kruzel; Editing by Kevin Liffey, Deepa Babington, David Gregorio and Sergio Non)