BEIRUT, Lebanon — Pitched battles raged between Israeli forces and Hezbollah fighters on the border yesterday, and Israel warned hundreds of thousands of people to flee southern Lebanon “immediately,”preparing for a likely ground offensive to set up a buffer zone.

As the death toll rose to 330 in Lebanon as well as at least 31 Israelis, Lebanese streamed north into the capital and other regions, crowding into schools, relatives’ homes, or hotels. Taxi drivers in the south were charging up to $400 a person for rides to Beirut — more than 40 times the usual price. In remote villages of the south, cut off by strikes, residents made their way out over the mountains by foot.

The price of food, medical supplies, and gasoline rose by as much as 500% in parts of Lebanon yesterday as Israel’s relentless bombardment destroyed roads, bridges, and other supply routes. The World Food Program said estimates of basic food supplies ranged from one to three months.

Hezbollah leader Sheik Hassan Nasrallah shrugged off concerns of a stepped-up Israeli onslaught, vowing never to release two Israeli soldiers captured by his guerrillas even “if the whole universe comes [against us].” He said they would be freed only as part of a prisoner exchange brokered through indirect negotiations.

He spoke in an interview with the Al-Jazeera news network taped yesterday to show he had survived a heavy airstrike in south Beirut that Israel said targeted a Hezbollah underground leadership bunker. The guerrillas said the strike only hit a mosque under construction and no one was hurt.

Israel has decided air power alone will not be enough to drive Hezbollah back from the Israeli-Lebanon border and that a ground force will be needed to establish a zone that is at least 20 miles deep, senior military officials said yesterday. That would force Hezbollah behind the Litani River.

Israel wants to send a strong message to all its enemies, especially Iran, that the consequences of attacking the Jewish state will be unbearable.

But mounting civilian casualties and the displacement of hundreds of thousands of Lebanese could limit the amount of time Israel has to achieve its goals, as international tolerance for the bloodshed and destruction runs out.

Prime Minister Siniora of Lebanon put the death toll at more than 330 — at least 11 of them killed yesterday — with 1,100 wounded. At least 31 Israelis have been killed, including 16 soldiers — two of them killed in yesterday’s fighting.

The United Nations estimated that about a half-million people have been displaced in Lebanon, with 130,000 fleeing to Syria and about 45,000 believed to be in need of assistance.

More than 600 relatives of U.N. peacekeepers and other foreigners were evacuated by ship from the southern port city of Tyre, a region south of the Litani that has seen a ferocious pounding by Israeli warplanes and gunboats for days. Many of the women and children had spent the night on the beach waiting for the ship that arrived yesterday morning and took them to Cyprus.

The exodus of Americans and other foreign nationals stepped up dramatically, with ships lining up off Beirut to take thousands of families waiting at the port out of the war zone.

Two large explosions shook south Beirut late yesterday in new Israeli strikes on the Shiite Muslim Hezbollah’s stronghold. During the day, Israeli strikes pounded villages and towns in the Shiite heartland of the south and the eastern Bekaa Valley. Hezbollah, in turn, fired more than 40 rockets into northern Israel.

The clashes about a mile inside the Lebanese side of the border yesterday evening came when an Israeli patrol sweeping for Hezbollah bunkers was ambushed by guerillas, taking casualties. The fight rapidly expanded, with Israeli helicopters firing missiles at targets on the ground and rescue force storming in.

The Israeli military said two Israeli soldiers died in the fighting and several guerrillas were killed. Hezbollah’s Al-Manar television said three Israeli soldiers were killed but did not mention guerrilla casualties.

Two Apache attack helicopters crashed in northern Israel near the Lebanon border early Friday, injuring four soldiers, the Israeli military said. Al-Jazeera reported that four soldiers were killed in the crash, but did not give a source.

Israel has stepped up its small-scale forays over the border in recent days, seeking Hezbollah positions, rocket stores, and bunkers. Each time it has faced tough resistance from the guerrillas.

In preparation for a more powerful punch deeper into Lebanon, an Israeli military radio station that broadcasts into the south issued what it called “a strict warning” that Israeli forces would “act immediately” to halt Hezbollah rocket fire.

“It will act in word and deed inside the villages of the south against these aggressive terrorist acts. Therefore all residents of south Lebanon south of the Litani must leave their areas immediately for their own safety,” the message in Arabic on the Al-Mashriq station said.

More than 300,000 people are believed to live south of the Litani — which twice has been the border line for Israeli buffer zones. In 1978, Israel invaded up to the Litani to drive back Palestinian Arab guerrillas, withdrawing from most of the south months later.