The Israeli cabinet has approved an army plan to push deeper into Lebanon, to try to take control of areas used by Hezbollah to launch rockets on Israel.
Up to 10,000 Israeli soldiers are already fighting in Lebanon
Up to 30,000 troops could take part, aiming to reach the Litani River, up to 30km (18 miles) inside Lebanon.
After nightfall, armoured columns begun moving into southern Lebanon under cover of intense artillery fire.
An Israeli military spokesman said the armed forces were operating according to the earlier political decisions.
Amid continuing attacks by Israel and Hezbollah, ceasefire moves were in jeopardy as US and France rowed over changes to a draft UN resolution.
Hezbollah's leader also attacked the current draft as "unjust".
Sheikh Hassan Nasrallah, in a television broadcast, said: "At the very least this resolution can be described as unfair and unjust. It gives Israel more than it wanted and more than it was seeking."
Attempts to reword the resolution, to take in Lebanese and Arab League demands for an immediate Israeli withdrawal, have so far proved impossible.
The BBC's Bridget Kendall says the mood at the UN is downcast, amid possible signs that both the French and Americans are beginning to worry the fragile diplomacy could collapse.
After Israel's decision to approve the new offensive, an Israeli spokesman told the BBC that Israel was waiting to see what progress, if any, was being made at the UN towards a ceasefire.
But our correspondent says the cabinet decision further darkened the mood at the UN.
The likely timetable for a vote - first mooted for Tuesday - appears to be slipping, possibly into next week. The White House said it had no idea when the Security Council would vote, AFP news agency said.
More than 1,000 people, most of them civilians, have now been killed in the conflict, the Lebanese government has said. More than 100 Israelis, most of them soldiers, have also been killed.
In other developments:
- Israel struck Lebanon's largest Palestinian refugee camp, Ein al-Hilweh, near the port city of Sidon, killing two people. The Israeli army said it was targeting a house belonging to a Hezbollah member
- Al-Arabiya TV reported that seven Israeli soldiers had been killed in a rocket attack in southern Lebanon - Israel confirmed two deaths in clashes with Hezbollah
- The death toll from an Israeli air strike on a residential building in southern Beirut on Monday went up to 41 after more bodies were found in the rubble
- The UN Human Rights Council will meet on Friday to discuss the conflict, after a request from 16 states led by Tunisia
The Israeli cabinet decision to push further into Lebanon came two days after Israel imposed an open-ended curfew on all residents south of the Litani River.
Beirut has been burying its dead after a major Israeli raid on Monday
Nine ministers voted in favour of the new offensive, and three abstained, amid reports that large Israeli casualties were feared.
One of the abstainers, Deputy Prime Minister Eli Yishai, said that weeks more conflict were anticipated.
"The assessment is it will last 30 days. I think it is wrong to make this assessment. I think it will take a lot longer," Mr Yishai said.
The 10,000 Israeli soldiers already in Lebanon are involved in fierce clashes with Hezbollah militants.
Sending in up to 30,000 troops is risky, says BBC security correspondent Frank Gardner in Jerusalem.
It presents Hezbollah with a large target - and if Israel plans to hold onto the territory, even for just 30 days, its troops will be vulnerable to ambush and its politicians will be accused of illegal occupation, he says.
Earlier, the Israeli military announced it was sending one of its most senior generals, Maj-Gen Moshe Kaplinsky, to co-ordinate the offensive.
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