The Palestinian death toll in an Israeli offensive in the Gaza Strip rose above 500 on Monday as the army said it had killed 10 militants who tunnelled into Israel, while Gazan officials said an Israeli tank shelled a hospital, killing civilians.

A day after he was caught by an open microphone saying sarcastically that the Israeli assault was "a hell of a pinpoint operation", US Secretary of State John Kerry flew to Cairo to try to secure an end to the two-week conflict.

Despite a UN Security Council appeal on Sunday for an immediate ceasefire in the worst bout of Palestinian-Israeli violence for more than five years, neither the Palestinian Islamist group Hamas nor Israel appeared ready to stop fighting.

Hamas, which killed 13 Israeli soldiers in Gaza on Sunday in the biggest one-day toll for eight years, continued to fire rockets deep into Israel and to dispatch infiltrators.

Israeli jets, tanks and artillery constantly pounded the densely-populated coastal strip, killing 28 members of a single family at the southern end.

At Al-Aqsa hospital in the central Gaza Strip, four people were killed and 70 wounded when an Israeli tank shell slammed into the third floor, housing operating theatres and an intensive care unit, the Health Ministry said.

The Israeli military, which has accused Hamas militants of firing rockets from the grounds of Gaza hospitals and seeking refuge there, had no immediate comment.

Non-stop attacks lifted the Palestinian death toll to 518, including almost 100 children, since fighting started on July 8, Gaza health officials said. Israel says 18 of its soldiers have also died along with two civilians.

Hamas announced late on Sunday it had captured an Israeli soldier in Gaza, displaying a photo ID card and serial number, but there was no confirmation from the Israeli side. The announcement set off rejoicing in the embattled Gaza Strip.

"This is not the time to talk of a ceasefire," 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 told reporters.

For its part, Hamas, weakened by the loss of Egypt and Syria as allies, voiced determination to fight on to break a blockade on Gaza imposed by both Israel and Egypt.

INFILTRATION ATTEMPT

Past conflicts between Israel and its foes in Gaza and Lebanon have usually ended when the United States, the Jewish state's guardian ally, calls a halt, sometimes hastened by a strike that inflicts high civilian casualties on the Arab side.

While Washington went along with Sunday's Security Council statement, it has so far defended Israeli actions and refrained from pressuring Netanyahu publicly to stop.

Violence along the Gaza border intensified on 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, the 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 army, 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 flying into the air.

A military spokeswoman said at least 10 militants died. She did not comment on reports of casualties amongst Israeli forces. Hamas said its men had 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 to destroy the tunnels and the militants' missile stock pile.  -Reuters