After meeting Palestinian Authority leaders in the West Bank, UN Secretary General Ban Ki-moon called on Israel to end the fighting.
Ban Ki-Moon: 'The fighting must stop'
"I would like to see an immediate ceasefire," he said, adding that a deal between Israel and Hamas to stop the conflict might be possible within the next few days.
But Israel insists any ceasefire has to be "durable and sustainable".
Hamas leader Khaled Meshaal said they would not accept Israeli conditions for a ceasefire.
Health officials in Hamas-controlled Gaza say at least 1,105 Palestinians have been killed and 5,100 wounded since Israel launched an operation on 27 December to end rocket attacks against its people.
Thirteen Israelis - three of them civilians - have died, while 233 soldiers have been wounded, the Israeli army says.
The BBC's Christian Fraser, allowed over the Egypt-Gaza border for the first time, says conditions for Palestinian families seeking refuge from the Israeli bombardment of Rafah in a UN-run school are very difficult.
There are hundreds of children there, he says, but food supplies are limited, electricity is available for only eight hours a day and there is no running water.
Our correspondent says that from the damage he has been allowed to see in Rafah, homes and a playground were right next to one Israeli strike on what it said were Hamas militants.
Israel has been bombing heavily along the border area, our correspondent says, with the aim of destroying tunnels running between Egypt and Gaza.
Tens of thousands of mourners took to the streets of Gaza City for the funeral of a top Hamas leader Said Siyam.
Mr Siyam, who controlled thousands of Hamas security troops in Gaza, died along with his son, brother and two other Hamas officials when his brother's house in Gaza City was bombed on Thursday.
Funeral procession for Hamas leader Said Siyam
Israeli military officials said 40 overnight air strikes targeted smuggling tunnels, rocket launching points, weapons stores and a militants' training camp.
The bodies of 23 people were later recovered in the Tel al-Hawa district of Gaza City, medics said.
The diplomacy now is in high gear. Hopefully we're entering the final act
In Gaza City, the BBC's Hamada Abuqammar says Israeli artillery fire has continued - even during the daily humanitarian ceasefire, although the bombardment was not as intense as on Thursday.
He said the Quds hospital was now empty, after sick and wounded patients had to be evacuated from the building overnight because of a fire caused by a tank shell.
The UN headquarters in Gaza was also hit on Thursday.
Hundreds of Gazans had been seeking refuge in the compound when it was attacked by Israeli shells, causing a fire that destroyed vital food supplies.
Speaking a day later, the senior UN official in Gaza, John Ging, described as "total nonsense" claims by the Israeli Prime Minister Ehud Olmert that militants were using the compound to fire on Israeli troops.
Meanwhile, rockets continue to be fired from Gaza into Israel. About 10 were launched on Friday but caused no injuries, the Israeli army said.
The violence is continuing amid a diplomatic push for a ceasefire.
This page is best viewed in an up-to-date web browser with style sheets (CSS) enabled. While you will be able to view the content of this page in your current browser, you will not be able to get the full visual experience. Please consider upgrading your browser software or enabling style sheets (CSS) if you are able to do so.