The new peace effort came as eight members of a single family were killed in Israeli tank fire in northern Gaza, after Israel warned it was poised to broaden its ground operations.
Earlier, US President Barack Obama said he had telephoned Israeli Prime Minister Benjamin Netanyahu to voice concerns about the crisis and Palestinian president Mahmud Abbas met foreign leaders for talks.
In the face of Israel's land, sea and air offensive, the Islamist movement Hamas remained defiant and warned the Jewish state it would "drown in the swamp of Gaza."
And battles continued early Saturday morning, with the army saying a firefight had caused "several injuries" to soldiers.
The UN said Secretary General Ban Ki-moon would fly to the region Saturday in a bid to end the violence.
"The secretary general... will leave for the region tomorrow to express solidarity with the Israelis and Palestinians," under secretary general for political affairs Jeffrey Feltman told emergency talks at the Security Council.
He said Ban would help Israelis and Palestinians "in coordination with regional and international actors, end the violence and find a way forward."
But the two sides' UN ambassadors traded blame for the violence, with Israel's Ron Prosor insisting no other country would "tolerate... terrorist" rocket fire at its citizens.
Palestinian ambassador Riyad Mansour read aloud the names of Palestinian dead, including women and children to the Security Council, and at one point appeared close to tears.
Israel's ground incursion, launched on the tenth day of an operation to stamp out rocket fire from Gaza, has killed dozens and forced thousands of people to flee.
The UN agency for Palestinian refugees UNRWA has opened 34 of its schools to shelter those fleeing.
It said the number of Gazans seeking sanctuary with the agency had more than doubled in one day from 22,000 to more than 47,000.
On Friday, Netanyahu said he had ordered the military to be ready for "a significant broadening of the ground activity."
He then convened his security cabinet to discuss a possible expansion of the campaign, which began on July 8 with the aim of stamping out cross-border rocket fire.
Israel has authorised the enlistment of 53,000 reserve troops, the army said.
- Eight family members killed -
In Gaza, after a relative lull during the day, violence picked up again in the evening, with intensifying tank shelling and air strikes killing more than a dozen people.
Among them were eight members of a single family killed by tank fire on their home in northern Gaza, emergency services spokesman Ashraf al-Qudra said.
Four children from the family were among the dead, joining another four children killed in several separate incidents of tank fire east of Gaza City, the youngest of them just two years old.
Those and other deaths raised the overall toll in 11 days of fighting in Gaza to 296 people killed.
An Israeli civilian and a soldier have also been killed.
In Gaza, the World Food Programme said it had already distributed emergency food rations and food vouchers to more than 20,000 displaced people.
But with the ground operation, it was gearing up for a huge increase in the coming days and hoping to reach 85,000 people with food distributions, a spokeswoman said.
Gaza was also struggling with a 70 percent power outage after electricity lines from Israel were damaged, officials said.
- Tunnel operation -
Israel has said the aim of the ground operation is to destroy Hamas's network of tunnels which are used for cross-border attacks on southern Israel.
Netanyahu said the ground operation was necessary to deal with the tunnels, but admitted there was "no guarantee of 100 percent success."
Obama told reporters the US supports Israel's right to defend itself, but said Washington was "deeply concerned about the risks of further escalation and the loss of more innocent life."
He added that Washington was "hopeful" that Israel would operate "in a way that minimises civilian casualties".
The EU called for an immediate ceasefire, reserving particular concern for "too many civilian deaths, including many children," and urged efforts towards a "lasting peace" between Israel and Palestinians.
Israel pulled out all of its troops and settlers from Gaza in 2005, but within a year it became the de facto seat of Hamas after it won a landslide victory in Palestinian parliamentary elections.
Meanwhile, Abbas arrived in Turkey where he urged support for an Egyptian-brokered ceasefire proposal.
French Foreign Minister Laurent Fabius said Abbas had urged Paris to ask Hamas allies Qatar and Turkey to pressure the group into accepting a ceasefire.