Israeli forces have launched limited incursions in Lebanon, the United States said on Monday, as Israel vowed to keep fighting Hezbollah and sealed part of the border after killing the Iran-backed militants' leader.

Israel's Defence Minister Yoav Gallant warned the battle was not over even after the massive Friday strike on Beirut that killed Hezbollah chief Hassan Nasrallah, dealing the group a seismic blow.

World leaders have urged diplomacy and de-escalation, with United Nations Secretary-General Antonio Guterres's spokesman Stephane Dujarric saying: "We do not want any sort of ground invasion."

Israeli officials "have informed us that they are currently conducting... limited operations targeting Hezbollah infrastructure near the border," US State Department spokesman Matthew Miller told journalists.

Hezbollah fighters were "ready if Israel decides to enter by land," deputy leader Naim Qassem said in a first televised address since Nasrallah's death.

Lebanon's national army, dwarfed by Hezbollah's military power, was "repositioning" troops farther from the border, a military official said.

And United Nations peacekeepers in Lebanon were no longer able to conduct patrols "given the intensity of the rockets going back and forth," Dujarric said.

US President Joe Biden, whose country is Israel's main weapons supplier, earlier on Monday indicated he opposes an Israeli ground operation.

"We should have a ceasefire now," he said.

Israel launched earlier this month a wave of deadly air strikes on Hezbollah strongholds across Lebanon, and on Friday bombed Nasrallah in Beirut.

In northern Israel, near the Lebanese border, Gallant said that "we will use all the means that may be required... from the air, from the sea, and on land" to restore calm.

He said the killing of Nasrallah "is an important step, but it is not the final one."

Lebanon's state-run National News Agency reported "heavy artillery shelling" at a border village in the country's south.

Hezbollah began low-intensity strikes on Israeli troops a day after its Palestinian ally Hamas staged its unprecedented attack on Israel on October 7 which triggered war in the Gaza Strip.

The border clashes rapidly escalated this month, leaving people across the region fearful of even more violence to come.

Israel said earlier this month that it was shifting its focus from Gaza to securing its northern border.

On Monday the army declared an area of the border strip a "closed military zone."

Israel's strikes on Lebanon have killed hundreds of people over the past week and forced up to a million to flee their homes, according to Lebanese officials.

Hezbollah and other groups launched rockets, drones and some missiles at Israel over the same period, causing some injuries but no deaths.

Israeli Prime Minister Benjamin Netanyahu accused arch-foe Iran, which backs Hamas, Hezbollah and other armed groups, of plunging "our region deeper... into war."

"There is nowhere in the Middle East Israel cannot reach," Netanyahu warned.

Iran has said Nasrallah's killing would bring about Israel's "destruction," though the foreign ministry said on Monday that Tehran would not deploy any fighters to confront Israel.

Lebanese Prime Minister Najib Mikati called for a ceasefire based on a recent US-French proposal, urging "an end to the Israeli aggression against Lebanon."

Most of Israel's strikes have targeted Hezbollah strongholds in eastern and southern Lebanon and the southern suburbs of Beirut, the group's main bastion.

Hamas said its leader in Lebanon, Fatah Sharif Abu al-Amine, was killed along with his wife and two children in a strike on Al-Bass refugee camp in south Lebanon. The Israeli military confirmed it had killed Sharif.

On Monday, an Israeli strike hit a building in central Beirut, with an armed Palestinian group saying it had killed three of its members.

The strike, the first in the city centre in years, sparked panic.

Central Beirut resident Kahier Bannout, 42, said it was "supposed to be a safe area -- not a war zone."

"Everyone is afraid."

Lebanon's Health Minister Firass Abiad said more than 1,000 people have been killed since September 17.

UN refugee agency chief Filippo Grandi said "well over 200,000 people are displaced inside Lebanon," while more than 100,000 have fled to neighbouring Syria.

European Union foreign policy chief Josep Borrell, after emergency talks between the bloc's top diplomats, warned that "any further military intervention will dramatically aggravate the situation and it has to be avoided."

"The sovereignty of both Israel and Lebanon has to be guaranteed," he said.

French Foreign Minister Jean-Noel Barrot, the first high-level diplomat to visit Beirut since the Israeli strikes intensified, on Monday urged Israel "to refrain from any ground incursion."

"There is still hope" for a ceasefire, he said, "but there is little time."

US Secretary of State Antony Blinken said diplomacy was the best path forward for the region.

Washington "will continue to work... to advance a diplomatic resolution" for the Israel-Lebanon border, and for a Gaza ceasefire and hostage-release deal, he said.

The US, Qatar and Egypt tried for months to broker such a deal, which Netanyahu's domestic critics accused him of obstructing.

In Gaza, witnesses said the number of Israeli air strikes has dropped significantly in recent days.

A UN Satellite Centre assessment issued on Monday said "two-thirds of the total structures in the Gaza Strip have sustained damage" in nearly a year of war.

With schools not operating and many of them destroyed, children in Gaza "are not allowed to have the life of normal children, no education, no play, no joy", Unicef spokesman for the Palestinian territories Jonathan Crickx said after a mission to Gaza.

Hamas's unprecedented October 7 attack on Israel resulted in the deaths of 1,205 people, mostly civilians, according to a tally based on Israeli official figures that include hostages killed in captivity.

Israel's retaliatory military offensive has killed at least 41,615 people in Gaza, most of them civilians, according to figures provided by the Hamas-run territory's health ministry. The UN has described the figures as reliable. (AFP)