* Hostilities subside ahead of Muslim Eid festival on Monday
* Israel says will only fire in response to attack
* Obama appears to link Hamas demilitarisation to peace (Adds Ban Ki-moon, Islamic Jihad comments, updates deaths, rockets)
By Nidal al-Mughrabi and Maayan Lubell
GAZA/JERUSALEM, July 28 (Reuters) - Israel eased its offensive in the Gaza Strip and Palestinian rocket fire from the enclave declined sharply on Monday, 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 respond to attacks for an indefinite period.
"The situation now is an unlimited truce," Israel's chief military spokesman, Brigadier General Motti Almoz, told Israel Radio. "The IDF (Israeli Defence Forces) is free to attack after any fire if there is any."
The Islamist Hamas movement which controls the Gaza Strip 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.
However, the lull appeared fragile amid diplomatic tension between Israel and its main sponsor, the United States.
U.N. Secretary-General Ban Ki-moon said parties to the fighting had "expressed serious interest" in his request for a further 24-hour humanitarian ceasefire, but "have not yet agreed on the timing of its implementation".
Israeli troops continued to hunt and destroy cross-border militant tunnels inside Gaza, and it was not clear if Hamas was ready to agree to a prolonged pause.
At least 12 rockets were fired out of the battered coastal territory at Israeli towns on Monday, according to the Israeli military, which said it struck two rocket launchers and a weapon manufacturing site in the northern and central Gaza strip.
Gaza residents reported Israeli shelling in east and northern Gaza and the health ministry said two people, including a five-year-old boy, were killed in one of the attacks. An army spokeswoman said she would look into the incident.
Jibril Jnaid, the boy's father, said his son, Sameeh, was playing with his cousins when a bullet fired from Israeli forces operating on the edge of Jabalya, the biggest refugee camp in northern Gaza, hit him and he fell bleeding to the ground.
"At the day of Eid, I am proud to sacrifice my son for the sake of victory of the resistance and the victory of the will of the Palestinian people," the father told Reuters as the boy's mother and other mourners kissed his cheeks before burial.
Hamas's armed wing said it killed two Israeli soldiers in the northern Gaza Strip. An Israeli military spokeswoman said a soldier was wounded there but she knew of no fatalities.
Some residents in Gaza reported they had received a recorded telephone message on Monday which said in Arabic: "Listen Hamas, if you are still alive, you should know that if you continue, we will respond, we will respond violently."
Israeli leaflets dropped over Gaza listed dozens of names of gunmen from Hamas and its ally, Islamic Jihad, that the military says it has killed since the start of the offensive.
"This list is part of the names of those who thought they could face the might of the Israeli Defense Forces," read the leaflet, which included a map to a graveyard where it said the militants were buried.
International efforts to secure a long-term ceasefire have so far faltered with Israel and Hamas presenting almost irreconcilable demands. The Gaza militants want an end to the Israeli-Egyptian blockade of their enclave.
Israel has said an easing of the siege would only come if the armed groups were stripped of their weapons. An opinion poll broadcast by Channel 10 tv showed overwhelming Israeli support for continuing the Gaza offensive until Hamas is "disarmed".
Deputy Islamic Jihad chief Zeyad Al-Nakhala said mediation had made progress and the group was working with Egypt to craft a deal.
"We are days away from the end of the battle, the clouds will clear and you (Palestinians) will see victory," he told Islamic Jihad's radio station Al-Quds, "We will not accept anything less than ending the blockade."
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.
After Israeli sources vented anger at Kerry's ideas, U.S. President Barack Obama telephoned Prime Minister Benjamin Netanyahu on Sunday to urge Israel hold fire unconditionally. The U.N. Security Council 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 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 1,037 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 between Netanyahu's government and Washington has flared over U.S. mediation efforts, adding yet another chapter to the prickly relations between the Israeli leader and Obama.
A U.S. official, speaking on condition of anonymity, challenged media leaks by unnamed Israeli officials damning as too accommodating to Hamas a draft accord attributed to Kerry. The official said 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 20 years have failed to broker a permanent deal. The most recent round 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.
Qatari Foreign Minister, Khaled Al-Atteya, told Al-Jazeera TV that Israel had not respected a ceasefire agreement that ended the last Gaza war in 2012 and it was time the blockade of Gaza was lifted.
"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 said.
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. (Additional reporting by Amena Bakr in Doha; Writing by Maayan Lubell and Dan Williams; Editing by Toby Chopra and Paul Taylor)