* Pre-dawn Israeli offensive is biggest in conflict so far
* Militants fire rockets at Israeli towns for fifth day
* Egypt leading diplomatic efforts to secure ceasefire
* U.S. envoy in region ahead of UN Security Council session (Edits, updates death tolls, adds White House, Golan Heights)
By Nidal al-Mughrabi and Stephen Farrell
GAZA/JERUSALEM, May 14 (Reuters) - As the war between Israel and Hamas militants in the Gaza Strip lit up the skies and sent civilians running for cover for a fifth night running on Friday, diplomats stepped up efforts to try and end the violence.
As Gazans marked a grim Eid al-Fitr feast marking the end of the holy month of Ramadan and Israel began a holiday weekend with no sign of an end to fighting, casualties spread further afield, with Palestinians reporting 11 killed in the West Bank amid clashes between protesters and Israeli security forces.
The Israeli military said more than 2,000 rockets had been fired from Gaza into Israel since the start of the conflict, around half of which were intercepted by missile defence systems and 350 fell into the Gaza Strip.
At least 126 people have been killed in Gaza since Monday, including 31 children and 20 women, and 950 others wounded, Palestinian medical officials said.
Among eight dead in Israel were a soldier patrolling the Gaza border and six Israeli civilians, including two children, Israeli authorities said.
Ahead of a session of the U.N. Security Council on Sunday to discuss the situation, Biden administration envoy Hady Amr, Deputy Assistant Secretary for Israel and Palestinian Affairs, flew in on Friday.
The U.S. Embassy in Israel said the aim of his trip was "to reinforce the need to work toward a sustainable calm."
Israel launched day-long attacks to destroy what it said were several km (miles) of tunnels, launch sites and weapon manufacturing warehouses used by the militants in an effort to halt the rocket attacks.
The Israeli operation included 160 aircraft as well as tanks and artillery firing from outside the Gaza Strip, Israeli military spokesman Lieutenant Colonel Jonathan Conricus said. It included a bank that, Israel said, handled the day-to-day cash flow of Hamas and its armed wing.
There have also been clashes between Jews and Israel's minority Arab community in mixed cities across Israel. The Israeli police said they had arrested at least 23 people over the unrest.
The Israeli army said late on Friday three rockets had been fired from Syria at the Israeli-controlled Golan Heights but there were no reports of any damage.
White House spokeswoman Jen Psaki said Washington was focused on leveraging U.S. relationships in the region to end the crisis diplomatically, adding that Israel had a right to self-defence.
Egypt was leading regional efforts to secure a ceasefire. Cairo was pushing for both sides to cease fire from midnight on Friday pending further negotiations, two Egyptian security sources said, with Egypt leaning on Hamas and others, including the United States, trying to reach an agreement with Israel.
The foreign ministers of Egypt and Jordan discussed efforts to end the Gaza confrontation and to prevent "provocations" in Jerusalem, Egypt's foreign ministry said.
"The talks have taken a real and serious path on Friday," a Palestinian official said. "The mediators from Egypt, Qatar and the United Nations are stepping up their contacts with all sides in a bid to restore calm, but a deal hasn't yet been reached."
U.N. Secretary-General Antonio Guterres also appealed for an immediate ceasefire.
"Fighting has the potential to unleash an uncontainable security and humanitarian crisis and to further foster extremism...," U.N. spokesperson Stephane Dujarric said.
Hamas, the Islamist group that rules Gaza, launched the rocket attacks on Monday, in retaliation for Israeli police clashes with Palestinians near al-Aqsa Mosque, Islam's third holiest site, in East Jerusalem.
Sheikh Ikrima Sabri, who led Friday prayers at al-Aqsa Mosque, decried its treatment by Israeli forces. He said its "sanctity has been violated several times during the holy month of Ramadan" in what he called violations "unprecedented" since the 1967 Arab-Israeli war.
Israel says it makes every effort to preserve civilian life, including warning in advance of attacks.
"What we were targeting is an elaborate system of tunnels that spans underneath Gaza, mostly in the north but not limited to, and is a network that the operatives of Hamas use in order to move, in order to hide, for cover," Conricus told foreign reporters, adding that the network was known as "the Metro".
The Israeli military has put the number of militants killed so far in the Israeli attacks at between 80 and 90.
The U.N. Office for the Coordination of Humanitarian Affairs said there were reports of more than 200 housing units destroyed or severely damaged in Gaza and hundreds of people seeking shelter in schools in the north of the coastal enclave.
Major airlines have suspended flights to Israel and at least two owners of tankers delivering crude oil asked to divert from Ashkelon to the port of Haifa, farther north of Gaza, because of the conflict, shipping sources said on Friday.
There were pro-Palestinian protests in Jordan, Bangladesh and elsewhere, but the broader picture across the Islamic world, where Muslims are marking the Eid al-Fitr holiday and where restrictions on movement due to COVID-19 are in place in some countries, was noticeably muted.
(Additional reporting by Rami Ayyub, Dan Williams and Ari Rabinovitch in Israel; Aidan Lewis in Cairo, Nandita Bose and Steve Holland in Washington, Michelle Nichols in New York and Emma Farge in Geneva Editing by Philippa Fletcher, Mark Heinrich, Frances Kerry and Gareth Jones)
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