Israeli ground forces advanced deep into Gaza’s most populated areas today, broadening their offensive even as diplomatic officials issued urgent appeals for a ceasefire.

Moving under the cover of night, Israeli soldiers edged into the city centres in the pre-dawn hours, engaging Palestinian militants in street battles in the fiercest fighting to date. Witnesses described how gunmen tried to lure Israeli soldiers into ambushes, as troops combed through Gaza’s southwestern suburbs and took up stronghold positions in outlying buildings.

Israel had dropped leaflets on Gaza Saturday warning of the offensive.

“The IDF (Israeli Defense Forces) is not working against the people of Gaza but against Hamas and the terrorists only,” the leaflets, in Arabic, explained. “Stay safe by following our orders.”

Ahmed Al-Attira, 57, said that his family is using the last of its food stock, but cannot leave the house for more supplies.

"We tried yesterday but there were attacks. Today it is even worse. They tell us that they are not working against us, but our people are dying and I do not know what I will do tomorrow. This thing is getting worse each day it continues."

There has been speculation that Israel would launch a third stage in its offensive this week, following a decision by the Israeli security cabinet to expand and broaden the operation.

A senior defence official described the third stage of the operation to The Times as a “comb-through” for rocket launchers and militants.

“We have captured the northern parts of Gaza that militants have used as their launching pad to bombard our cities with rockets. They are still firing from within the city centres.They are putting their own civilians at risk, and trying to draw us in, but we must protect our citizens."

The first stage of the Israeli operation was a massive aerial bombardment of Hamas targets, while a second stage launched a ground invasion of the coastal strip.

Ehud Olmert, the Israeli prime minister, said that Israel would continue its offensive until it achieved it’s primary goal of quelling all rocket attacks.

“Israel is getting close to achieving the goals it set. But patience, determination and effort are still needed to realise these goals in a manner that will change the security situation in the south," Olmert said.

Since Israel began its offensive 16 days ago, Gaza Health officials have counted more than 800 dead, at least half of them civilians. The Israeli military said that troops had killed some 300 armed militants. 13 Israelis have died, three of them civilians.

Both sides have ignored calls by the UN Security Council for an immediate ceasefire, calling the resolution “unworkable.”

Israel’s conditions for a ceasefire include a complete stop to Hamas rocket-fire and weapons smuggling from Egypt. Hamas is demanding that Israel open all border crossings with Gaza.

Egyptian officials have sought to mediate between the two sides, which refuse to talk to each other directly. A Hamas delegation is now in Cairo, while Israel plans to send senior defence official Amos Gilad on Monday.

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