Palestinian mourners pray in a mosque during the funeral for those killed in a three-storey house belonging to the Abu Jamaa family the day before, in Khan Yunis in the southern Gaza Strip, on July 21, 2014. Mohammed Abed / AFP / Getty Images

Palestinian health official Ashraf al-Kidra said the shells killed at least four people and wounded 60, including 30 medical staff, at Al Aqsa hospital in the town of Deir el-Balah. He said the shells landed in the administration building, the intensive care unit and the surgery department. The Israeli military said it was looking into the report.

Both sides ignored a Sunday U.N. Security Council appeal for an immediate cease-fire. The Israeli military on Monday said it killed 10 Palestinian fighters who slipped across the border from Gaza through hidden tunnels, while an Israeli airstrike hit a Palestinian hospital.

U.S. Secretary of State John Kerry headed to Egypt on Monday as part of efforts to halt the bloodshed as President Barack Obama reiterated his call for an immediate cessation of hostilities. Meanwhile, U.N. Secretary General Ban Ki-moon was touring the Middle East trying to build momentum for a cease-fire to end a conflict that has killed 550 Palestinians and 27 Israelis, 25 of whom were soldiers.

International pressure for an end to the fighting in Gaza grew Monday after the bloodiest day in Israel’s weeks-long offensive on the besieged enclave — a bombardment that has left more than 500 dead, most of them civilians.

It comes amid a worrying surge in civilian deaths after the launch of a ground offensive Thursday. Reuters, citing medics on the ground, reported that Israeli shelling late on Sunday killed 28 members of a single family near the Gaza’s southern border with Egypt.

Hamas’ armed wing and other Palestinian groups continued to fire missiles across southern and central Israel on Monday, and heavy fighting was reported in the north and east of Gaza.

Meanwhile, the fate of an Israeli soldier reportedly captured by Palestinian fighters Sunday remains uncertain. Hamas announced late on Sunday that the man had been taken, setting off celebrations in Gaza. It named the soldier as Shaul Aron and showed his ID papers but did not release a picture of him alive in their hands.

Israel's ambassador to the United Nations denied that a soldier was in captivity. But the Israel's military said it was still investigating. "We still cannot rule it out," military spokesman Lt. Col. Peter Lerner said on Monday.

Amid the fighting, hundreds of thousands of Gazans have been told to leave their homes. According to the United Nations Relief and Works Agency, which is responsible for the welfare for Palestinian refugees, the conflict has displaced 85,000 Gazans, who are now living in 67 shelters.

Despite worldwide calls for a cessation of the worst bout of Palestinian-Israeli violence for more than five years, Israeli ministers have seemingly ruled out any swift truce.

"This is not the time to talk of a cease-fire," said Gilad Erdan, communications minister and a member of Israeli Prime Minister Benjamin Netanyahu's inner security cabinet. "We must complete the mission, and the mission cannot end until the threat of the tunnels is removed," he said.

But international pressure for a cease-fire appears to be growing. And Washington public support for Israel’s right to defend itself seems to be coupled with growing concern over casualties.

On Sunday, Kerry seemed to voice his displeasure at the toll Israel's operation was taking on Palestinian civilians in private comments to an aide caught by a live microphone by Fox News.

His visit to Cairo as part of peace efforts were not accompanied by any discernable lightening of the cross-border barrage between Gaza and Israel.

Violence along the border intensified Monday, and sirens wailed across much of central and southern Israel to warn of rocket attacks. At least nine missiles were shot down by Israel's Iron Dome interceptor, its army said.

Looking to take the fight onto Israeli soil, two groups of Palestinian fighters crossed from Gaza via two tunnels in the early morning, opening fire as they entered.

Black and white surveillance footage supplied by the Israel Defense Forces showed one group of five or six men crouching and firing in long grass. Seconds later, they were hit by a large explosion, which sent a cloud of smoke and debris into the air.

An Israeli military spokeswoman said at least 10 Palestinian fighters died. She did not comment on reports of casualties among Israeli forces. Hamas said its men destroyed an army jeep in the assault.

Fighters from Hamas, which controls Gaza, and its allies have repeatedly tried to infiltrate Israel over the past week through a vast network of hidden tunnels, looking to attack villages and army encampments that dot the border area.

Netanyahu sent in Israeli ground forces on Thursday with the express aim of destroying the tunnels and the militants' missile stockpile.

Lerner told reporters that the main focus of fighting remained the Shujayea district, east of Gaza City, where some 72 Palestinians, many of them civilians, were killed on Sunday.

The push in Shujayea also saw Israel suffer its worst losses in the offensive, with 13 soldiers killed — the army's heaviest one-day loss in battle since 2006.

Al Jazeera and wire services