Israel Pushes Deeper Into Gaza

Despite Olmert's strong rhetoric in Ashkelon, he appeared open to securing a truce with Hamas through Egypt.

"I really hope that the efforts we are making with the Egyptians these days will ripen to a result that will enable us to end the fighting," Olmert said, according to the AP.

Israeli officials claim that there is growing discord within Hamas on the direction of the conflict. GOC Intelligence Major General Amos Yedlin says the Hamas leadership in Damascus, Syria, has lost control of the situation in Gaza.

Yuval Diskin, the director of the Israeli security service, Shin Bet, who briefed ministers at Sunday's cabinet meeting claims Hamas in Gaza is eager for a cease-fire, and that he believed Hamas' popularity there was lessening.

"The leadership in Gaza understands that compromise is inevitable," said Yedlin during the cabinet meeting.

In an e-mail message today, Hamas leader Ismail Radwan said his group would not consider a cease-fire before Israel stops its attacks and pulls back from Gaza, and he demanded opening of all border crossings, according to The Associated Press, which also reported that British Foreign Secretary David Miliband said today that Egyptian negotiators have indicated Hamas is showing some urgency in its desire for a cease-fire deal to end the fighting in Gaza.

Hamas leaders in the Gaza Strip, however, declared victory to be "closer than ever," according to Reuters.

Sources inside the Israeli government tell ABC News there is a split is widening between the country's top leaders, with Defense Minster Ehud Barak and Foreign Minister Tzipi Livni opting for a unilateral withdrawal, while Olmert prefers increasing the intensity of fighting.

Olmert would still need approval from the Cabinet before the offensive could be pushed into a third phase -- in which the army takes over larger areas of Gaza.

Today thick, black smoke could be seen rising over the eastern Gaza suburbs, where opposing forces skirmished throughout the night. At least 20 Palestinians were killed in new air strikes or died from their wounds today, including two women and two children, Gaza health officials said.