Iran and the United States agreed Monday to set up communications lines to keep the vital Strait of Hormuz open and end fighting in Lebanon, mediators said, after their first round of talks in Switzerland toward ending the war in the Middle East.

The teams led by U.S. Vice President JD Vance and Iran's Mohammad Bagher Ghalibaf opened talks Sunday, as part of a two-month negotiating period set out under a preliminary deal agreed last week.

Some south Lebanon residents cautiously headed back to their hometowns on Sunday after fighting paused between Israel and Hezbollah the previous evening, even as Lebanon's army warned locals to delay returns.

Israel's military chief said on Sunday that Hezbollah had suffered a severe blow fighting Israeli forces and was now in a "very difficult position", as he met with troops in southern Lebanon.

He spoke as the United States and Iran held talks in Switzerland after signing a preliminary agreement to end the broader Middle East war, with the conflict in Lebanon threatening to derail the deal.

Israeli Prime Minister Benjamin Netanyahu said on Sunday that Israeli forces would remain in southern Lebanon "as long as necessary", while also vowing to prevent Iran from acquiring nuclear weapons.

"We will remain in the security zone in southern Lebanon for as long as necessary to protect the cherished residents of the north and all the citizens of Israel... Nothing will alter that commitment," Netanyahu said.

Hezbollah chief Sheikh Naim Qassem on Sunday rejected any Israeli security zone in Lebanon, after Israel's Prime Minister Benjamin Netanyahu vowed his troops would remain in the country's south for as long as necessary.

Israeli troops "remaining on Lebanese land is impossible. There are no security zones for Israel... we have a national army which deploys, and it is responsible for preserving sovereignty, and it is who we cooperate with," Qassem said in a televised address, adding that "Israel is an aggressor and must leave."

Iranian state television reported on Sunday a pause in the U.S.-Iran talks mediated by Pakistan and Qatar in Switzerland, and said that the Islamic republic's nuclear program had not been discussed.

"No negotiations regarding Iran's nuclear program took place during the 80-minute first round of talks," the state broadcaster said, adding that the focus had been on the implementation of the memorandum of understanding between Tehran and Washington, and the situation in Lebanon.

Iran's chief negotiator Mohammad Bagher Ghalibaf on Sunday warned the United States against making threats at the Islamic republic, vowing that "our armed forces are ready to respond".

U.S. President Donald Trump said Sunday in a surprising statement that "Iran must immediately stop their highly paid PROXIES in Lebanon from causing trouble," apparently referring to Iran-backed Hezbollah.

Iran said on Sunday that the ongoing conflict in Lebanon between Israel and Hezbollah will top the agenda in talks with the United States in Switzerland, as well as issues such as frozen Iranian funds and the sale of the country's oil.

The Israeli military said that one of its soldiers was killed in fighting in southern Lebanon on Saturday, marking the fifth such fatality since a U.S.-Iran deal was reached to end the Middle East war.