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The US is not actively seeking a ceasefire deal between Israel and HezbollahÌýbutÌýinstead trying to limit Israeli military operations in Lebanon, US officials have told CNN.

A senior US official, speaking on condition of anonymity, told CNN that although talks will take place with Israelis and Lebanese to de-escalate the situation in Lebanon, Israel had "to get beyond some things" before the prospect of ceasefire can be discussed.

It comes amid a massive build-up of Israeli troops on the southern Lebanon border and an intensive air campaign which has killed senior Hezbollah commanders as well as many civilians.

"We couldn't stop them from taking action, but we can at least try to shape what it looks like," the official said.

White House Press Secretary Karine Jean-Pierre confirmed that talks regarding a diplomatic solution for Lebanon were taking place but did not elaborate.

"We're going to be having regular consultations with the Israelis, with the Lebanese and others, regarding the right moment to press for such an agreement," said Jean-Pierre on Monday.

French President Emmanuel Macron and his US counterpart Joe Biden last month proposed a 21-day truce in Lebanon to give time for negotiations for a longer-lasting ceasefire deal with Israel.

The call was backed by several US and French allies and came only two days after Israel began a deadly aerial campaign in Lebanon, carrying out massive airstrikes and killing hundreds.

But Israeli Prime Minister Benjamin Netanyahu refused the proposal, insisting on continuing efforts to destroy Hezbollah’s military capabilities amid the killing of Hassan Nasrallah and other figures.

The Israeli military is seeking to expand its invasion of southern Lebanon and continue striking the area, claiming that it wants to clear the region of anything elements that threaten its security.

The officials told CNN there are rising concerns in Washington that what Israel vowed would be a "limited" ground operation in southern Lebanon will grow into a large-scale and prolonged war.

Hezbollah is a proscribed terrorist organisation in the US and many allied states. While Washington supports Israel targeting the Iran-backed group's leaders and infrastructure, it has called on Israel to avoid large civilian casualties.

Hundreds of civilians have been killed in Israel’s massive aerial offensive in the country’s south and east since 23 September, while more than 1.2 million have been displaced.

The Lebanese health ministry put the death toll at 2,083 on Monday, including hundreds of Hezbollah fighters, after a whole year of cross-border fighting. Most civilian deaths were recorded since Israel began escalations more than two weeks ago, among them children, rescuers, and paramedics.

Israel ramps up ground offensive

The Israeli military said it was deploying a fourth division to its northern front. It announced that its 146th Reserve Division began incursions in the western part of south Lebanon Monday night.

This adds thousands of Israeli troops to the already more than 15,000 deployed inside south Lebanon, according to The Times of Israel.

Israel claims it has managed to destroy a large part of the group’s arsenal in airstrikes in recent weeks.

The Israeli military Tuesday renewed evacuation orders for anybody still present south of the Litani River -Ìýwhich Israel wants as the border for a buffer zone between the two warring parties- warning them they were risking their lives if they returned home as Israel continues to "target Hezbollah."

On Monday, the Israeli military told fishermen and beachgoers to clear the southern Lebanon coast, signalling a wider offensive.

Hezbollah commander killed

Hezbollah says it has successfully repelled Israeli forces from advancing further into Lebanese territory since 1 October, claiming to have killed or injured dozens of Israeli soldiers.

The Lebanese Shia movement is reeling under recent severe blows by Israel with thousands of its fighters maimed after their communication devices were remotely detonated, and most of its senior command structure killed, including Nasrallah.

The fate of Hashem Safieddine, Nasrallah’s cousin who was expected to take over as Hezbollah’s secretary-general, is still unknown after a massive airstrike last week in a south of Beirut suburb.

In the latest assassination, Israel claimed Suhail Hussein Husseini, who commanded the group’s headquarters south of Beirut, was killed on Tuesday.

Husseini, who was a member of Hezbollah's Jihad Council, participated in weapon transfers between the group and Iran and oversaw the distribution of advanced weaponry among Hezbollah units.

Despite these setbacks, Hezbollah had vowed to continue the fight against Israel and the groupÌýfired more rockets into northern Israel on Monday and Tuesday, reaching Tel Aviv and Tiberias.

Over the weekend, Hezbollah missiles hit the centre of Haifa for the first time since fighting began last year, causing injuries. Hezbollah said it targeted a number of military bases and military gatherings.

Hezbollah will continue to fight

Hezbollah deputy leader Naim Qassem on Tuesday played down talk that his group was weak or could not continue to confront Israel.

"Everyone is trying to pressure us so we can be frightened, but we tell them we are not scared, we are the children of Nasrallah," Qassem, who seemed to be in a poorly lit room with only curtains seen behind him, said in a televised speech.

"The [Israeli] occupation’s aim is to end the resistance entirely so that there is no more resistance against Israel, and to genocide the Palestinian people so that they are incapable of creating a future resistance.

"The party's leadership and the resistance (Hezbollah) are meticulously organised... we have overcome painful blows. I would like to reassure you that our capabilities are fine."

He defended Hezbollah's support for Gaza a year ago, saying it aimed at helping the Palestinians there achieve victory and that Lebanon was inevitably going to be the next target of an Israeli assault.

"We will strike Israel at the time and place we see is right," he said. "Netanyahu will not achieve his goals".

Qassem also thanked Iran’s support for the group, saying the country "supported and gave Hezbollah a lot over the years".

Despite the language, the deputy leader did say he supported Lebanese efforts for a ceasefire.

"We support the political efforts that [Lebanese Parliament Speaker] Nabih Berri is undertaking towards a ceasefire," Qassem said in a televised speech.

"Once a ceasefire is achieved, diplomacy can look into all the other details."

After his speech, aroundÌý by Hezbollah towards the northern Israeli cities of Haifa and Acre.

Hezbollah has received backlash from Lebanese rivals about opening a front against Israel and warned Hezbollah for months about risking a wider war in Lebanon, already mired by its worst-ever economic crisis.