Israel has pummelled Gaza for a 14th day, raising the Palestinian death toll to more than 580, as ceasefire efforts continue in Doha and Cairo.
With air strikes and shelling raining down across the besieged coastal enclave, Israel's army on Tuesday said two of its soldiers had been killed the day earlier, bringing the Israeli toll to 27 troops and two civilians in the bloodiest Gaza conflict since 2009.
Washington urged Israel to take "greater steps" to prevent innocent casualties, and UN chief Ban Ki-moon appealed for the violence to "stop now" as it emerged most of the dead were civilians.
Hamas insisted on a lifting of Israel's siege of Gaza and the release of prisoners in order to agree to any ceasefire accord.
US Secretary of State John Kerry pledged $47m in humanitarian aid to Gaza as he called for an end to the violence.
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"We are deeply concerned about the consequences of Israel's appropriate and legitimate effort to defend itself," he told reporters as he met with the UN chief.
Kerry, Ban, and Arab League chief Nabil al-Arabi urged Hamas to accept the Egyptian-proposed ceasefire.
Meanwhile, Hamas leader Khaled Meshaal and Palestinian president, Mahmoud Abbas, held talks in Doha, the Qatari capital, pledging to work together for a ceasefire and to lift the blockade on Gaza.
Hamas reiterated the necessity on a lifting of Israel's blockade of Gaza and the release of prisoners to halt its rocket fire.
"The conditions for a ceasefire are ... a full lifting of the blockade and then the release of those recently detained in the West Bank," its leader in Gaza, Ismail Haniya, said on television.
"We cannot go backwards, to a slow death," he said, referring to the Israeli blockade in force since 2006.
Monday's attacks across Gaza killed at least 56 people including 16 children, bringing the overall death toll since Israel launched its operation on July 8 to 583 Palestinians, according to figures provided by Gaza emergency services spokesman Ashraf al-Qudra. More than 3,600 Palestinians have been injured in the attacks.
In the costliest single Israeli bombardment, an air strike hit a residential tower block in central Gaza City, killing 11 people, including five children.
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It came after Israeli tank shells struck a hospital in Deir al-Balah, killing at least five people, including doctors, officials said, indicating at least 70 people were wounded.
Rights group Amnesty International warned Monday that the shelling of the hospital as well as the "continuing bombardment of civilian homes" in Gaza, "add to the list of possible war crimes that demand an urgent independent international investigation".
Since the Israeli operation began on July 8, huge numbers of Palestinians have fled their homes, with the UN saying more than 100,000 people have sought shelter in 69 schools run by its Palestinian refugee agency (UNRWA).
Deadly clashes in southern Israel
Violence also broke out in the West Bank, where an Israeli shot dead a Palestinian who had been throwing stones at his car, Palestinian security sources said.
An Israeli army spokeswoman said the military was investigating the death, which she said took place during "a violent riot".
Israeli police said Palestinians had rioted in East Jerusalem neighbourhoods Monday night, with no casualties.
Elsewhere in the West Bank, an Israeli was wounded after a Palestinian opened fire at him from a travelling car, the army said.
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On Monday, Israeli forces killed more than 10 fighters who had infiltrated southern Israel, the army said, later announcing it had lost four soldiers in that battle.
The troops who died in that clash were among seven killed in 24 hours, said the Israeli army, adding 30 soldiers were wounded over the same period. Two soldiers were also killed in the Gaza Strip, the army said on Tuesday.
That brought its toll to 27 soldiers killed since the start of the operation, including 13 on Sunday, the bloodiest single day for the Israeli military since the Lebanon war of 2006.
At least 116 rockets hit Israel on Monday, one striking the greater Tel Aviv area, and another 17 were shot down, the Israeli army said.