SIDON, Lebanon – Mideast diplomats were pressing Syria to stop backing Hezbollah as the guerrillas fired more deadly rockets onto Israel’s third-largest city Sunday. Israel faced tougher-than-expected ground battles and bombarded targets in southern Lebanon, hitting a convoy of refugees.

Israel’s defense minister said his country would accept an international force, preferably NATO, on its border after it weakens Hezbollah in southern Lebanon. But his troops reported encountering an intelligent, well-prepared and ruthless guerrilla army whose fighters don’t seem to fear death.

With Secretary of State Condoleezza Rice arriving in Israel on Monday, both the Arabs and Israelis appeared to be trying to set out positions ahead of Washington’s first diplomatic mission to the region since the fighting began. The United States backs Israel’s refusal to talk about a cease-fire until it completes the military campaign against Hezbollah, but is under increasing pressure to foster a plan to end the growing suffering and destruction in Lebanon.

Still, daily casualty figures appeared to be lowering – about nine confirmed Sunday by Lebanese security officials, compared with dozens each day earlier in the week. The decrease could be a result of the exodus from the hardest-hit areas or because of the difficulty for authorities in getting figures from the war zone.

In the 12th day of fighting, guerrillas launched a new barrage of more than a dozen rockets against the Israeli city of Haifa, killing two people and setting an apartment building on fire. Israeli missiles struck a convoy of fleeing Lebanese, killing four people, including a journalist.

In the far south, fighting with Hezbollah raged around the Israeli military’s foothold in Lebanon – the border village of Maroun al-Ras, where the Israeli army has maintained a significant presence since Saturday. But so far, they were not advancing. Hezbollah reported three of its fighters killed.

With Israel and the United States saying a real cease-fire is not possible until Hezbollah is reined in, Arab heavyweights Egypt and Saudi Arabia were pushing Syria to end its support for the guerrillas, Arab diplomats in Cairo said.

A loss of Syria’s support would deeply weaken Hezbollah, though its other ally, Iran, gives it a large part of its money and weapons. The two moderate Arab governments were prepared to spend heavily from Egypt’s political capital in the region and Saudi Arabia’s vast financial reserves to break Damascus from the guerrillas and Iran, the diplomats said.