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Ten civilians and two field commanders from the Islamic Jihad faction were killed and at least 30 other Palestinians were hurt in the new air strikes, hospital officials said, bringing the death toll from six days of clashes in Gaza to 85.
United Nations Secretary General Ban Ki-moon was due to arrive in Cairo to weigh in on ceasefire efforts led by Egypt, which borders both Israel and Gaza and whose Islamist-rooted government has been hosting leaders of Hamas.
Israeli media said a delegation from Israel had also been to Cairo for truce talks, though a spokesman for Prime Minister Benjamin Netanyahu's government declined comment on the matter.
The Gaza flare-up, and Israel's signalling that it could soon escalate from the aerial bombings to a ground sweep of the cramped and impoverished enclave, have stoked the worries of world powers watching an already combustible region.
As Hamas and other Islamist factions spurn permanent peace with the Jewish state, mediated deals for each to hold fire unilaterally have been the only formula for stemming bloodshed in the past. But each side now placed the onus on the other.
Izzat Risheq, aide to Hamas politburo chief Khaled Meshaal, wrote on Facebook that Hamas would enter a truce only after Israel "stops its aggression, ends its policy of targeted assassinations and lifts the blockade of Gaza".
Listing Israel's terms, Vice Prime Minister Moshe Yaalon wrote on Twitter: "If there is quiet in the south and no rockets and missiles are fired at Israel's citizens, nor terrorist attacks engineered from the Gaza Strip, we will not attack."
Yaalon also said Israel wanted an end to Gaza guerrilla activity in the neighbouring Egyptian Sinai, a desert peninsula where lawlessness has spread during Cairo's political crises.
Israel's operation has so far drawn Western support for what U.S. and European leaders have called its right to self-defence in the face of years of cross-border attacks, but there have also been growing appeals for an end to the hostilities.
Sympathy for Israel may wear thin as the Gaza toll mounts. On Sunday (local time), 11 Palestinian civilians were apparently killed during an Israeli attack on a militant which brought a three-storey family home crashing down on them.
"I am deeply saddened by the reported deaths of more than ten members of the Dalu family... (and) by the continuing firing of rockets against Israeli towns, which have killed several Israeli civilians. I strongly urge the parties to co-operate with all efforts led by Egypt to reach an immediate ceasefire," Ban said before leaving for Egypt. He visits Israel on Tuesday.
At least 22 of the Gaza fatalities have been children.
Netanyahu said he had assured world leaders that Israel was doing its utmost to avoid causing civilian casualties in Gaza.
In scenes recalling Israel's 2008-2009 winter invasion of Gaza, tanks, artillery and infantry have massed in field encampments along the sandy, fenced-off border and military convoys moved on roads in the area.
Israel has also authorised the call-up of 75,000 military reservists, so far mobilising around half that number.
A big, bloody rocket strike on Israelis might be enough for Netanyahu to give a green light for a ground offensive.
Three Israelis have been killed and dozens wounded in hundreds of salvoes since Wednesday. Some rockets reached as far as Tel Aviv, Israel's commercial capital, but were shot down by the country's air defence system.
As a precaution against the rocket interceptions endangering nearby Ben-Gurion International Airport, civil aviation authorities said on Monday new flight paths were being used. There was no indication takeoffs and landings at Ben-Gurion had been affected.
There was no rocket fire from Gaza between midnight and daybreak on Monday, the Israeli military said. It said a few cross-border launches followed in the early morning but there was no immediate word of casualties in southern Israel, where such salvoes usually set off sirens so residents can shelter.
Israel bombed some 80 sites in Gaza overnight, the military said, adding in a statement that targets included "under-ground rocket launching sites, terror tunnels and training bases" as well as "buildings owned by senior terrorist operatives".
Israel's declared goal is to deplete Gaza arsenals and force Hamas to stop rocket fire that has bedevilled Israeli border towns for years. The rockets now have greater range, putting Tel Aviv and Jerusalem within their reach - a strategic weapon for Gaza's otherwise massively outgunned guerrillas.
The southern resort city of Eilat was apparently added to the list of targets when residents said they heard explosions on Sunday and Monday thought to be rockets, though there was no word of casualties or damage.
Eilat is thought to be well out of the range of any rocket in possession of Hamas or any other Gaza group. But militants have in the recent past fired rockets at Eilat and its surroundings, using Egypt's Sinai desert as a launch site.
Hamas and other groups in Gaza are sworn enemies of the Jewish state which they refuse to recognise and seek to eradicate, claiming all Israeli territory as rightfully theirs.
Hamas won legislative elections in the Palestinian Territories in 2006 but a year later, after the collapse of a unity government under President Mahmoud Abbas the Islamist group seized control of Gaza in a brief and bloody civil war with forces loyal to Abbas.
Abbas then dismissed the Hamas government led by the group's leader Ismail Haniyeh but he refuses to recognise Abbas' authority and runs Gazan affairs.
While it is denounced as a terrorist organisation in the West, Hamas enjoys widespread support in the Arab world, where Islamist parties are on the rise.
U.S.-backed Abbas and Fatah hold sway in the Israeli-occupied West Bank from their seat of government in the town of Ramallah. The Palestinians seek to establish an independent state in the West Bank and the Gaza Strip with East Jerusalem as its capital.