The fighting came after US President Donald Trump declared Monday that he had brokered a deal which the Lebanese embassy in Washington said would at first only cover Israeli attacks on Beirut and Hezbollah attacks on Israeli territory, before expanding in scope.

Israel has been fighting Hezbollah since the group dragged Lebanon into the wider Middle East war by attacking Israel on March 2 in support of Iran.

Neither side has publicly accepted Trump's deal, with senior Hezbollah official Mahmud Qomati telling AFP in a written statement the group "will not accept a partial ceasefire".

Lebanon's state-run National News Agency reported Israeli strikes, some of them deadly, on around 30 locations across the south on Tuesday.

Hezbollah meanwhile said it had attacked Israeli troops in southern Lebanese lands they occupy, but had not claimed attacks in Israel.

The Israeli military said it intercepted two projectiles from Lebanon, without reporting any injuries.

The fighting took place with Israeli and Lebanese diplomats in Washington for a fourth round of direct talks since the start of the current war.

"Israel and Lebanon can do a peace deal tomorrow," Rubio told a hearing of the US Senate's Foreign Relations Committee.

He added: "Israel has no territorial claims in Lebanon. Hezbollah is the impediment."

Lebanese Prime Minister Nawaf Salam called the talks, which are strongly opposed by Hezbollah, "the least costly choice for Lebanon".

- 'Crazy' -

Rubio said Washington wanted the talks to remain independent of those with Iran to end the wider Middle East war launched by the US and Israel against Tehran on February 28.

Tehran has repeatedly linked the two conflicts and on Monday said that Israel's expanding campaign in Lebanon risked ending the US-Iran ceasefire in place since April 8.

Recent days have seen a dramatic escalation in fighting and bombardment as Israeli troops staged their deepest ground offensive into Lebanon in two decades.

Citing what he called Hezbollah's "repeated violations" of a ceasefire officially in place since April 17 but never respected by either side, Israel's Prime Minister Benjamin Netanyahu ordered strikes on Beirut's southern suburbs, a densely populated Hezbollah stronghold.

According to US site Axios, however, Trump pressured Netanyahu to back down, calling him "crazy" in a phone call and accusing him of putting peace talks with Iran at risk.

Israeli Defence Minister Israel Katz subsequently said that Israel had established "a new equation" backed by Washington that his country would hit the Beirut suburbs if Hezbollah continued firing at Israel.

In the southern suburbs, which many residents had fled the day before, many shops were closed Tuesday, while a military drone flew over the area at low altitude, according to an AFP journalist.

Resident Layla Shehab, 35, said she had decided to return as "we found the situation has calmed down a bit".

Citing Israel's actions in Lebanon, Iran's Tasnim news agency reported on Monday that Iran was suspending peace talks with Washington.

Trump on Tuesday denied the report, however, insisting that the US and Iran were speaking "continuously" including "one day ago and today".

- Shelters full -

Near Sidon, in southern Lebanon, rescuers recovered the bodies of six members of the same family following an Israeli strike.

Further south in the historic city of Tyre, the Jabal Amel hospital resumed operations after being severely damaged on Monday.

Lebanon's health ministry said Israeli strikes in the south killed five people on Tuesday and wounded 48, including a doctor and five employees of Tebnine Governmental Hospital, which sustained damage.

The Israeli military meanwhile released a statement alleging Hezbollah members were operating in Tyre's Christian quarter -- so far spared from evacuation warnings and strikes -- saying it would order people to leave should the group remain there.

A few thousand people remain in Tyre's small old city, where the Christian quarter is located.

With shelters full, displaced residents were sleeping in cars or tents.

An AFP correspondent said some people started leaving the area after the Israeli military's statement.

Lebanon's health ministry said on Tuesday that Israeli attacks had killed more than 3,465 people since March 2.

At least 26 Israeli soldiers and one civilian contractor have been killed in the same period.