EIRUT (AFP) - US Secretary of State Condoleezza Rice is in Beirut at the start of a high-stakes mission to try to end the devastating conflict in Lebanon, even as Israeli troops were locked in deadly combat with Hezbollah guerrillas.

Two soldiers were reported killed in pitched battles with Shiite Muslim militiamen on the 13th day of cross-border warfare as Israel troops in tanks, bulldozers and armoured personnel carriers pushed even deeper into Lebanon on Monday.

Rice, whose government has been steadfast in its support for Israel's fearsome war on Hezbollah, made a surprise visit to Beirut in what US officials said was a "show of support" for Lebanon.

"We believe that a ceasefire is urgent," Rice told reporters en route to the region, where she is also due to hold talks with Israeli Prime Minister Ehud Olmert and then Palestinian leader Mahmud Abbas.

"It is however important to have conditions that make it sustainable."

The visit came as Washington appeared increasingly estranged from European and Arab allies over Israel's massive onslaught that has set off fears of a humanitarian catastrophe as thousands of foreigners and Lebanese flee.

At least 363 people have been killed in Lebanon, along with 37 confirmed Israelis, in the deadliest cross-border conflict since the Jewish state invaded its northern neighbour in 1982, triggering accusations that it was using excessive force to stop Hezbollah.

Israel, which is struggling to knock out Hezbollah despite its vastly superior military might, has suggested it would accept some form of international peacekeeping force in southern Lebanon.

Despite the stepped-up diplomacy, fighting continued as Israeli forces moved towards Bint Jbeil, the largest town in the border zone and a Hezbollah stronghold, after taking control of the nearby strategic village of Marun al-Ras.

Al-Jazeera television reported that two soldiers were killed but this was not confirmed by Israel.

"Heavy exchanges of fire, with automatic weapons and anti-tank rockets, is taking place this morning between Israeli soldiers advancing from Marun al-Ras toward Bint Jbeil... and Hezbollah combattants," a local police officer said.

A combat helicopter also crashed in northern Israel near the border with Lebanon, causing victims, the army said. Hezbollah claimed it had shot it down.

The army also said two Hezbollah militiamen captured by troops near Marun al-Ras on Sunday had been brought to Israel.

Washington has come under increasing pressure for bold action amid criticism it is stalling to allow Israel time to attempt to wipe out Hezbollah, which set off the conflict after seizing two Israeli soldiers on July 12.

But Hezbollah chief Hassan Nasrallah remained defiant, vowing that deeper incursions would not stop the rocket fire, and ruling out any efforts for a negotiated settlement unless it involved a prisoner swap.

"We are truly in a state of war and Hezbollah's priority is to stop the savage Zionist aggression on Lebanon," he told As-Safir newspaper. "We do not feel that we are currently interested in discussing ideas or initiatives."

One person was killed in a new round of air strikes on Monday on southern Lebanon, which has borne the brunt of the bombardments.

Streams of people, many waving white flags, have been making a desperate trek from the area after Israel ordered them to leave their homes, raising fears it was planning a large-scale ground invasion.

And as the grim death toll mounts, so has international criticism of the offensive, which has left Lebanon virtually cut off from the world, made hundreds of thousands of people refugees in their own country and destroyed billions of dollars of infrastructure.

UN humanitarian coordinator Jan Egeland, in Lebanon to launch a urgent appeal for funds for half a million people made homeless by Israel's onslaught, criticised both Israel and Hezbollah for attacking civilians.

"My position is very clear -- the hostilities must stop immediately. Civilian populations are not targets. That is against the law, humanitarian law," he said on a visit to a refugee centre in the mountains east of Beirut.

Even a minister from close US ally Britain, which had drawn Arab anger for appearing to back US support of the bombardment, has described Israel's tactics as "very difficult to understand".

Israel's ambassador to the United States rejected the criticism, saying the assault had dealt a "real blow" to Hezbollah, damaging the group's arsenal and killing a "few hundred" of its fighters.

But in the first openly expressed reservations by an Israeli minister on the success of the offensive, minister without portfolio Eitan Cabel said: "I admit I had hoped for better from the army."

General Udi Adam, the commander of Israel's northern military region, said the offensive will continue for "several weeks" although it now appears Israel has backed off from a full-scale invasion.

Before her departure from the United States, Rice and President George W. Bush met visiting Saudi Foreign Minister Prince Saud al-Faisal, who was also reported to have presented concrete plans for a truce.

Saudi officials said he proposed an exchange of prisoners between Hezbollah and Israel, something the Israeli government has so far ruled out without the prior release of the two soldiers.

Syria, which backs Hezbollah and is blamed by the United States for stoking the conflict, has warned that if Israel invaded Lebanon it would have no choice but to respond.

"What will we do? Stand by with our arms folded? Absolutely not. Without any doubt Syria will intervene in the conflict," Information Minister Moshen Bilal told a Spanish newspaper.

US ambassador to the United Nations John Bolton rebuffed a previous Syrian offer of dialogue in characteristically blunt fashion, saying "Syria doesn't need dialogue to know what they need to do."

Olmert, whose army is also fighting a second deadly offensive in the Gaza Strip that has left over 100 Palestinians dead, said he would accept a peacekeeping force in Lebanon "made up of troops from European Union countries".

Its mandate "will have to include control of the border crossings between Syria and Lebanon, deployment in south Lebanon and support for the Lebanese army," he said.

As the bombardments continue across Lebanon, foreign governments have been forced to lay on a flotilla of ferries, warships and cruise liners to evacuate stranded nationals, mainly to the nearby resort island of Cyprus which has been battling to find temporary accommodation and flights for the estimated 70,000 evacuees at peak summer holiday season.