Israel's growing offensive in Lebanon, including heavy bombardment and its deepest ground invasion in two decades, has threatened to scupper a ceasefire in the wider Middle East war between the United States and Iran.

Hezbollah drew Lebanon into the conflict on March 2 by firing rockets at Israel in retaliation for the US-Israeli killing of Iran's supreme leader, and Tehran has demanded that Lebanon be included in any peace deal with Washington.

The overnight clashes came after Lebanon said Iran-backed Hezbollah had accepted a US proposal for a "mutual cessation of attacks" after Trump said he persuaded the warring sides to de-escalate.

Under the arrangement, Israeli strikes on Dahiyeh, Beirut's southern suburbs, would cease in exchange for Hezbollah refraining from launching attacks against Israel, according to a statement from the Lebanese embassy in Washington.

But Israeli Prime Minister Benjamin Netanyahu appeared to cast doubt on any truce, and a report by US news outlet Axios on Monday said Trump called him "fucking crazy" and accused him of putting Iran peace talks at risk.

"There will be no Troops going to Beirut, and any Troops that are on their way, have already been turned back," Trump said on his Truth Social network after a "very productive" call with Netanyahu.

"Likewise, through highly placed Representatives, I had a very good call with Hezbollah, and they agreed that all shooting will stop -- That Israel will not attack them, and they will not attack Israel," he added.

Netanyahu later said he had told Trump "that if Hezbollah does not cease attacking our towns and our citizens, Israel will strike terrorist targets in Beirut".

But Trump doubled down on his ceasefire push, posting on Truth Social that "hopefully" Israel and Hezbollah would stop fighting "for ETERNITY!"

- Trading blows -

The announcements came on the eve of a fourth round of US-hosted direct negotiations between Israel and Lebanon on Tuesday and Wednesday. Military delegations held security talks last week.

But Hezbollah claimed multiple attacks on Israeli targets, mainly in south Lebanon, including after Trump's announcement.

Hezbollah also claimed a rocket attack on an Israeli tank early Tuesday in Hadatha, in southern Lebanon, saying on Telegram it was fighting "the advance of Israeli forces".

Lebanon's state-run National News Agency (NNA) reported late Monday that Israeli strikes on several areas in the south including in the villages of Marwaniyeh, Sidiqine, Yater and Mansouri, and also said "a very violent detonation" rocked the town of Debbine.

The Israeli military, for its part, said it intercepted two projectiles from Lebanon, without any casualties on Tuesday.

Tehran has said a Lebanon ceasefire remained a key condition for any deal with Washington.

Near the border, Father Antonios Farah, a priest for the village Qlayaa, where some residents were still holding out despite Israeli evacuation orders, told AFP a strike on a car had killed a man, his son and his daughter, who were students, as he drove them back from university exams in Beirut.

Lebanon's health ministry said Israeli attacks since March 2 have killed at least 3,433 people.

Israel's military said two of its soldiers had been killed in southern Lebanon, bringing to 27 the number of Israeli military deaths since early March.

- 'Fears intensified' -

Stephane Dujarric, spokesman for UN Secretary-General Antonio Guterres, urged all sides "to respect the cessation of hostilities".

In a report to the UN Security Council seen by AFP, Guterres said it would be necessary to maintain peacekeepers in Lebanon after the mandate of the current mission expires at year-end.

Israel's military has urged residents of southern Beirut, a Hezbollah stronghold mostly spared heavy attacks since April, to evacuate.

AFP journalists saw families fleeing the densely populated southern suburbs.

South Beirut resident Hadi, 24, said he had hoped for some stability during the truce, but "that feeling did not last long".

"Our fears intensified this morning" after the Israeli announcement, which "caused widespread panic, and we immediately left the area", he told AFP by telephone.

A truce to halt the fighting in Lebanon began on April 17, but has never been observed. Both Israel and Hezbollah accuse each other daily of violating the ceasefire, justifying their attacks by blaming the other's alleged breaches.