Egypt's President Mohammed Morsi said Israel's "aggression" against Gaza will end on Tuesday and Cairo-mediated truce efforts will produce results within hours, the official MENA news agency reported.
"The farce of the Israeli aggression will end today, Tuesday, and the efforts to reach a ceasefire between the Palestinians and Israelis will produce positive results within a few hours," it quoted him as saying.
A Hamas official said chief Khaled Meshaal and his negotiators were currently in a meeting with the Egyptian intelligence chief mediating the talks. "But it's no secret we're on the verge of an agreement," he said.
He said Hamas was still insisting Israel lift its six-year blockade of the Palestinian enclave for a truce, although it was not clear whether the Jewish state has agreed to this.
Israel on Thursday halted a threatened Gaza ground offensive to give the truce talks a chance, after an overnight meeting of senior Israeli ministers weighed the Egyptian proposal.
Even as the Hamas official in Cairo was speaking, Israel dropped leaflets ordering Gaza City residents to evacuate immediately. Earlier, a rocket fired from Gaza struck just south of Jerusalem.
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"For your own safety, you are required to immediately evacuate your homes and move toward Gaza City centre," the Arabic-language leaflet said, specifying certain roads that should be used.
Morsi's announcement comes amid intense diplomatic efforts that brought UN Secretary General Ban Ki-moon on a visit to Cairo on Monday and Israel on Tuesday to bolster efforts to end the week-long conflict.
US Secretary of State Hillary Clinton, on a Southeast Asia tour with President Barack Obama, was also travelling to the region to meet leaders including Israeli Prime Minister Benjamin Netanyahu.
"The best way to solve this is through diplomacy, so that you have a peaceful settlement that ends that rocket fire and allows for a broader calm in the region," senior Obama aide Ben Rhodes said.
Arab countries meanwhile ratcheted pressure on Israel by sending a delegation of foreign ministers headed by Arab League chief Nabil al-Arabi to Gaza earlier on Tuesday.
An overnight lull in the fighting was broken during the morning by new Israel air strikes on Gaza City and the north that killed three people, the Hamas-run emergency services said.
At least 116 Palestinians and three Israelis have been killed in the conflict as Israel carried out air strikes and naval bombardment of Gaza.
Since the violence erupted on November 14 with an Israeli targeted killing of a top Hamas military commander, militants have fired more than 1,000 rockets at the Jewish state, killing three people and injuring dozens.