Israeli soldiers fought heavy battles with militants early on Tuesday as ground troops on Gaza City's outskirts pushed deeper into a southern neighbourhood.
Israeli fighter jets and helicopters also carried out dozens more attacks during the night, most of them as they were providing air support to the ground troops against Palestinian militants confronting them, a military spokesman in Tel Aviv said.
A total of 60 targets were struck, including more smuggling tunnels and a hotel in the north of the Strip in which militants had gathered and were planning to carry out a shooting against the ground troops, he said.
The fighting around Gaza City intensified, as international efforts to reach a diplomatic solution to the Gaza crisis continued.
The UN Security Council - whose resolution of Thursday calling for an immediate ceasefire has thus far not been heeded - was scheduled to meet again in New York Tuesday.
Secretary General Ban Ki-moon was also set to travel to the region Tuesday for direct talks with the parties. He was scheduled to make stops in Egypt, Jordan, Israel, the West Bank, Lebanon, Syria and Kuwait.
Ismail Haniya of Hamas, the de-facto prime minister in Gaza, emerged in a pre-recorded televised address from his hide-out for the second time since Israel's Gaza offensive began Dec 27, saying his movement was willing to cooperate on any initiative that would "end the aggression against our people".
He mentioned an Israeli withdrawal and the full opening of border crossings as conditions. Gaza will not break and "victory" is near, Haniya vowed.
His movement and other militant factions continued their attacks against Israel, firing some 20 rockets on Monday. The Palestinian toll of the offensive - launched to end more than seven years of such rocket and mortar attacks from Gaza at southern Israel - stands at at least 919 killed and more than 4,300 wounded. Thirteen Israelis have been killed.