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Extremist Israeli ministers have called on their military to invade and occupy south Lebanon and destroy the Hezbollah armed group, as cross-border fighting escalates.

On Monday, Israel’s National Security Minister Itamar Ben-Gvir said Israel should storm Lebanon and "destroy Hezbollah in its entirety."

Ben-Gvir echoed similar threats made a day earlier by Israel’s finance minister, saying that while he agreed with Bezalel Smotrich’s call to forcefully push Hezbollah away from the border and set up a buffer zone in southern Lebanon, such a plan "wouldn’t be enough."

"We would need to do another thing — and that’s a war that would destroy our enemies," Ben-Gvir said about Hezbollah.

"Even if they’re pushed back from the border, even if there’s a security zone, even if they move a bit — you can’t just leave people behind whose entire purpose and essence is to destroy the State of Israel," the far-right minister said.

"What they don’t do in six months they will do in a year, and what they don’t do in a year they will do in two," he added. "We must not leave this to our children. Not in the south [Gaza Strip] and not in the north [Lebanon]."

During a Sunday meeting of his extremist Religious Zionism party in northern Israel, Smotrich said a public ultimatum must be issued to Hezbollah to completely stop firing missiles and drones at Israel and withdraw all forces beyond Lebanon’s Litani River.

"If the ultimatum is not fully met, the IDF (Israeli military) will launch an assault deep in Lebanese territory to defend the northern communities, including ground entry and Israeli military takeover of the southern Lebanese area," warned Smotrich, who also serves in the Israeli defence ministry.

Since October 8 last year, Lebanon’s Iran-backed Hezbollah Shia group has traded fire with the Israeli military, resulting in the deaths of over 300 people in Lebanon – mostly fighters – and over a dozen Israelis.

Hezbollah says the fighting is a distraction tactic aimed at turning some of Israel’s attention away from the war-battered Gaza Strip, where over 35,000 people have been killed by aÌýrelentless and indiscriminate Israeli offensive.

Tens of thousands of residents of southern London have been forced to flee their homes, weighing heavily on Beirut, already mired in a deep, years-long economic crisis.Ìý

Israelis have also fled their homes in northern Israel andÌýBenjamin Netanyahu'sÌýwar cabinet is facing increasing pressure from Israelis for decisive action against Hezbollah.

Ben-Gvir and Smotrich are among those pressuring Netanyahu to push forward with the Gaza war and launch a full-scale attack on Lebanon.

"The way to bring the [evacuated] residents home to the north is through a military decision with a devastating assault on Hezbollah, its infrastructure and the destruction of its power," Smotrich said Sunday.

US and French-led mediation efforts are seeking to end hostilities between Hezbollah and Israel and implement a UN Security Council Resolution which ended the 2006 summer war between the two sides.

The deal would also finalise the border demarcation between the two countries with Israel still occupying some Lebanese land.

Hezbollah has refused to move its fighters away from the border and refuses a ceasefire before the devastating war on Gaza ends. Israel has repeatedly threatened to wage a full-scale war in Lebanon if diplomatic efforts fail.