Israel eased its assaults in the Gaza Strip and Palestinian rocket fire from the enclave declined sharply on Monday, the military said, with both the United States and United Nations calling for a durable ceasefire.

As international pressure mounted to end a 21-day conflict in which more than 1,000 people have been killed, an Israeli military official said the army would only fire in response to attacks, adding that this would be for an "unlimited" period.

However, Israeli troops continued to hunt and destroy cross-border militant tunnels inside Gaza, and it was not clear if Hamas Islamists who control the small enclave were ready to agree to a prolonged pause.

Hamas said on Sunday it wanted a 24-hour truce to mark the Muslim holiday of Eid al-Fitr, which started on Monday. In the hours after its announcement, Gaza gradually fell quiet.

Just one rocket was fired out of the battered coastal territory at the Israeli city of Ashkelon in the first nine hours of Monday, the military said. Gaza residents reported brief bursts of tank shelling and no casualties.

"This ceasefire or abatement is dynamic on the ground. If we need to, we will respond," Israel's chief military spokesman, Brigadier General Motti Almoz, told local media.

U.S. President Barack Obama on Sunday urged Prime Minister Benjamin Netanyahu to hold fire unconditionally, while the U.N. Security Council agreed a statement that called on both sides to implement a humanitarian truce that stretched beyond Eid.

Netanyahu's security cabinet met into the early hours of Monday to debate ceasefire proposals and also a possible escalation of the Gaza offensive, which Israel says was needed to halt Hamas rocket fire and destroy its tunnel network.

Israeli air, sea and ground attacks have killed some 1,031 Palestinians, mainly civilians and including many children, Gaza officials say. Israel says 43 of its soldiers have died, along with three civilians killed by rocket and mortar fire from Gaza.

TENSION

Tension between Netanyahu's government and Washington has flared over U.S. mediation efforts to end the three-week-old war, adding yet another chapter to the prickly relations between the Israeli leader and Obama.

U.S. Secretary of State John Kerry visited the region last week to try to stem the bloodshed, his contacts with Hamas - which Washington formally shuns - facilitated by Egypt, Turkey, Qatar and Western-backed Palestinian President Mahmoud Abbas.

Israel wants Egypt, which also borders the Gaza Strip, to take the lead in curbing the Palestinian Islamists. It worries about Doha and Ankara championing Hamas demands to open up the blockaded territory.

A flurry of media leaks by unnamed Israeli officials damning a draft agreement attributed to Kerry as too accommodating of Hamas was challenged by a U.S. official who, also anonymously, told reporters U.S. efforts had been mischaracterised.

Obama appeared to link Israel's core demand for Hamas to be stripped of cross-border rockets and infiltration tunnels, to a peace accord with the Palestinians that is nowhere on the diplomatic horizon.

Repeated U.S.-led negotiations over the past 20 years have failed to broker a permanent deal. The most recent round of talks collapsed in April, with Palestinians livid over Jewish settlement building in the occupied West Bank and Israelis furious that Abbas had signed a unity pact with old foe Hamas.

"The President stressed the U.S. view that, ultimately, any lasting solution to the Israeli-Palestinian conflict must ensure the disarmament of terrorist groups and the demilitarization of Gaza," the White House said.

It added that while Obama wanted any truce to be along the lines of an Egyptian deal that ended the last Gaza war, in November 2012, the United States also supported "regional and international coordination to end hostilities".

Qatari Foreign Minister, Khaled Al-Atteya, told Al-Jazeera TV that Israel had not respected the 2012 ceasefire agreement and said that it was time the blockade of Gaza was lifted.

"The demands of Palestinian brothers are fair and they are the minimum demands for a dignified life," he said.

"We have worked with the U.S secretary of state and we were about to achieve substantial results, and the brothers in Hamas acted positively, but the one who rejected the Kerry proposal was Israel," he added.

Speaking on Sunday, Netanyahu sounded open to easing conditions for the Gaza Strip's 1.8 million Palestinians but said this must be "intertwined" with disarming Hamas.

"I think you can't get social and economic relief for the people of Gaza without having an assured demilitarisation," he told CNN.

Israel says the Palestinians have lost around half of their rockets during the fighting - an account disputed by Hamas - and that army engineers have located and destroyed many of the tunnels from the territory. Those excavations will continue under any short-term truce, Israel says.

The main U.N. agency in Gaza, UNRWA, said more than 167,000 displaced Palestinians had taken shelter in its schools and buildings, following repeated calls by Israel for civilians to evacuate whole neighbourhoods ahead of military operations.

The Gaza turmoil has stoked tensions amongst Palestinians in mainly Arab East Jerusalem and in the occupied West Bank, which Abbas governs in uneasy coordination with the Israelis.

Medics said eight Palestinians were killed on Friday in incidents near the West Bank cities of Nablus and Hebron - the sort of death toll reminiscent of previous anti-Israel revolts.

*This story was edited by Ahram Online.