* Obama appears to link Hamas demilitarisation to peace
* Hostilities subside ahead of Muslim festival on Monday
By Nidal al-Mughrabi and Dan Williams
GAZA/JERUSALEM, July 28 (Reuters) - Israel sees no need for another Gaza ceasefire, an Israeli official was quoted as saying on Monday, as tensions between Prime Minister Benjamin Netanyahu's government and Washington flared over U.S. mediation to end the almost three-week-old war.
Fighting had subsided over the weekend, with the battered Palestinian enclave's dominant Hamas Islamists endorsing a U.N. call for a 24-hour halt ahead of Monday's Eid al-Fitr festival.
Yet Israel balked, having abandoned its own offer to extend a 12-hour truce from Saturday as Palestinian rocket launches persisted. Netanyahu's security cabinet met into the early hours of Monday to debate proposals including for an escalation of the Gaza offensive in which almost 1,100 people have died.
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 and views Hamas as a security threat, 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 the top diplomat's efforts had been mischaracterised.
But U.S. President Barack Obama, phoning Netanyahu on Sunday, put pressure on Israel to hold fire unconditionally and appeared to link its 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.
"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".
Israel did not immediately respond nor publish what, if anything, was decided at the overnight security cabinet session.
But Israel Radio quoted an unidentified government official as saying: "There is no need for any more ceasefires. Let Hamas stop firing first."
ISRAEL LINKS GAZA RELIEF TO DISARMING HAMAS
That signaled preference for a de facto mutual halt to fighting rather than any agreement preserving Hamas's arsenals and shoring up its status by improving Gaza's crippled economy.
Two decades of Israeli-Palestinian peace negotiations have made little progress and been frequently interrupted, most recently in April when Netanyahu called off talks overseen by Kerry in response to Abbas's surprise power-share with Hamas.
Speaking earlier 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.
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.
A poll published by Israel's Channel 10 television on Sunday said some 87 percent of respondents wanted Israel to continue the operation until Hamas was toppled. Another poll, published in the Jerusalem Post newspaper, found that 86.5 percent of Israel's majority Jews opposed calling a truce while rocket fire continued and Gaza retained any of the cross-border tunnels.
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 most 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 167,269 displaced Palestinians have taken shelter in its schools and buildings, following repeated calls by Israel for civilians to evacuate whole neighbourhoods ahead of military operations.
But residents of villages near the southern town of Khan Younis on Sunday attacked offices of the International Committee of the Red Cross, torching furniture and causing damage. They said the organisation had not done enough to help them.
During the lull in fighting inside Gaza on Sunday, residents flooded into the streets to discover scenes of massive destruction in some areas, including Beit Hanoun in the north and Shejaia in the east.
An Israeli official said the army hoped the widespread desolation would persuade Gazans to put pressure on Hamas to stop the fighting for fear of yet more devastation.
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. (Writing by Dan Williams; Editing by Paul Simao)