GAZA CITY, Gaza Strip — The U.N. chief and the U.S. secretary of state headed to Cairo on Monday to try to end two weeks of Israel-Hamas fighting that has killed at least 508 Palestinians and 20 Israelis and displaced tens of thousands of Gaza residents.

The new cease-fire efforts by U.N. Secretary-General Ban Ki-moon and U.S. Secretary of State John Kerry followed the deadliest day of fighting since the escalation erupted on July 8.

In New York, the U.N. Security Council expressed “serious concern” about Gaza’s rising civilian death toll and demanded an immediate end to the fighting following an emergency session.

As Israeli airstrikes continued to pound Gaza, rescue workers near the southern Gaza city of Khan Younis were digging out bodies early Monday from the one-story home of the Abu Jamea family, flattened in one of the strikes overnight, said Ashraf al-Kidra, a Gaza health ministry official.

Al-Kidra said the Palestinian death toll from the two-week offensive stood at 508 as of Monday morning. More than half of those victims — 268 — were killed since an Israeli ground operation in Gaza began late Thursday.

That total included 20 bodies that were found at the site near Khan Younis, where two people were pulled alive from the rubble, Al-Kidra said.

Elsewhere in Gaza, he said, Israeli tanks opened fire on the home of the Siyam family west of Rafah in the southern part of the strip, killing 10 people, including four young children and a 9-month-old baby girl.

“Without any warning at all they began bombarding us at midnight, at 2 a.m., said Dr. Mahmoud Siyam, the head of the family. “We are not related to any military or political activities. We are civilized people (living) in this area of Gaza, what crime have we committed?”

Meanwhile, the Israeli military said it foiled a Hamas infiltration attempt on Monday through two tunnels leading from northern Gaza into southern Israel. The military said 10 infiltrators were killed after being detected and targeted by Israeli aircraft.

On Sunday, the first major ground battle in two weeks of Israel-Hamas fighting exacted a steep price, killing 65 Palestinians and 13 Israeli soldiers and forcing thousands of terrified Palestinian civilians to flee their devastated Shijaiyah neighborhood, which Israel says is a major source for rocket fire against its civilians.

Large sections of Shijaiyah were pulverized by a barrage of Israeli tank and artillery bombardments and repeated Israeli air strikes that buffeted the densely populated neighborhood for most of Sunday.

Speaking on national television shortly after the military announced the deaths of the 13 Israeli soldiers,  Prime Minister Benjamin Netanyahu said the Gaza offensive would continue “as long as necessary” to end attacks from Gaza on Israeli civilians.

Appearing with Netanyahu, Defense Minister Moshe Yaalon said that Israel expected to complete its work neutralizing the Hamas tunnels leading into Israeli territory within several days — a possible hint of a timeframe for the end of the operation.

Still, much work remains if diplomats are to succeed in brokering a sustainable cease-fire. On Sunday, Kerry said the U.S. still supports the Egyptian proposal for a halt to the hostilities that Israel accepted and Hamas rejected last week.

Hamas remains deeply suspicious of the motives of the Egyptian government, which has banned the Muslim Brotherhood, a group that Hamas closely identifies with.

The 13 Israeli soldiers who died in Shijaiyah brought the overall Israeli death toll to 20, including two civilians who died from rocket and mortar fire directed at Israeli towns and villages from different parts of Gaza.

On Sunday evening, Hamas spokesman Mushir al-Masri in Gaza claimed his group had captured an Israeli soldier. An announcement on Gaza TV of the soldier’s capture set off celebration in the streets of West Bank.

But there was no official confirmation of the claim in Israel. Earlier, the Israeli ambassador to the U.N., Ron Prosor, said the Hamas claim was untrue.

For Israelis, a captured soldier would be a nightmare scenario. Hamas-allied militants seized an Israeli soldier in a cross-border raid in 2006 and held him captive in Gaza until Israel traded more than 1,000 Palestinian prisoners, some of whom were involved in grisly killings, for his return in 2011.