Across Gaza, Israeli fighter planes hit homes and a high-rise tower, burying families in the rubble. The strike on the Gaza City tower brought down most of the building, killing 11 people - including six members of the same family - and wounding 40, said Palestinian health official Ashraf al-Kidra.
Israeli tanks, meanwhile, shelled a hospital in central Gaza, killing four people and wounding dozens as the daily death toll surpassed 100 for a second day. Israel said the shelling targeted rockets hidden near the compound, and accused militants of using civilians as shields.
At least 565 Palestinians have been killed and more than 3,600 wounded in the past two weeks, al-Kidra said.
On the Israeli side, seven more soldiers were killed in clashes with Gaza fighters Monday, bringing the military death toll to 25 - more than twice as many as in Israel's last Gaza ground war in 2009.
Two civilians have also died in Palestinian rocket attacks on Israeli cities and scores of soldiers have been injured.
The mounting bloodshed brought U.N. Secretary-General Ban Ki-moon and U.S. Secretary of State John Kerry to Cairo on Monday, for a new cease-fire push. However, the gaps remain wide and no credible mediator has emerged.
Egypt, Israel and the U.S. back an unconditional cease-fire, to be followed by talks on a possible new border arrangement for Gaza. Israel and Egypt have severely restricted movement in and out of Gaza since Hamas seized the territory in 2007.
Hamas, with some support from Qatar and Turkey, wants guarantees on lifting the blockade before halting fire. The Islamic militant group has no faith in mediation by Egypt's rulers, who deposed a Hamas-friendly government in Cairo a year ago and tightened restrictions on Gaza - to the point of driving Hamas into its worst financial crisis since its founding in 1987.
The top Hamas leader in Gaza, Ismail Haniyeh, said Monday that Gaza's 1.7 million people share Hamas' goal of forcing Israel and Egypt to lift the blockade.
"We cannot go back, we cannot go back to the silent death" of the blockade, he said. "Gaza has decided to end the blockade by its blood and by its courage."
After a meeting with Egyptian Foreign Minister Sameh Shukri, Ban said that "violence must stop by all sides," and that they must enter negotiations, seemingly siding with Cairo's approach.
However, Ban also said: "We can't claim victory simply by returning matters to where they stood before they led to terrible bloodshed."
The border blockade has set Gaza back years, wiping out tens of thousands of jobs through bans on most exports and on imports of vital construction materials Israel says could be diverted by Hamas for military use.
Israel allows many consumer goods into Gaza, but experts say Gaza's economy cannot recover without a resumption of exports.
The Rafah passenger crossing with Egypt is Gaza's only gate to the world, but Egypt has tightened restrictions over the past year, allowing only medical patients, Muslim pilgrims and Gazans with foreign passports to travel.
On Monday, President Barack Obama reaffirmed his belief that Israel has the right to defend itself against rockets being launched by Hamas into Israel. Yet, he contended that Israel's military action in Gaza had already done "significant damage" to the Hamas terrorist infrastructure and said he doesn't want to see more civilians getting killed.
As Obama spoke, Kerry flew to Cairo to join diplomatic efforts to resume a truce that last had been agreed to in November 2012.
Kerry headed almost immediately into a meeting with Ban, where he announced the U.S. will send $47 million in humanitarian aid for tens of thousands of Palestinians who have fled their homes in Gaza to escape the violence. Kerry's top aides warned, however, that achieving an immediate and lasting cease-fire would be difficult and he hoped to make any progress over the next several days to secure even a temporary pause in the bloodshed.
It's not clear exactly what Israel and Hamas would each demand in return for agreeing to a truce now, but senior State Department officials said the issue of opening border crossings - potentially into Israel and Egypt - was under discussion.
"We will work to see if there is some way to not only arrive at a cease-fire of some kind but to get to a discussion about the underlying issues," Kerry said at the start of his meeting with Ban. "Nothing will be resolved by any cease-fire, temporary or long, without really getting to those issues at some point and that's what we need to do."
Kerry is expected to meet with top Egyptian officials, including President Abdel Fattah al-Sisi, the foreign minister Shukri, and Arab League President Nabil Elaraby over the next few days. But there were no immediate plans for face-to-face meetings with officials from Qatar, Turkey, Israel and the West Bank, and the State Department aides said it remained uncertain what could be accomplished in the talks.
Israeli Prime Minister Benjamin Netanyahu told Sky News Arabia that the goal of Israel's strikes on Gaza is "to restore quiet and security for the people of Israel for a significant period of time."
"We'll take whatever action is necessary to achieve that goal," he said, adding that Israel has accepted cease-fire offers, while Hamas has rejected them.
Israel has said it is trying to minimize civilian deaths and has accused Hamas of using civilians as human shields.
However, Israeli strikes on homes have driven up casualties.
About half of the dead were killed in their homes, according to the Al Mezan Centre for Human Rights, a Palestinian human rights group. Various Palestinian rights groups estimated that at least 425 homes across Gaza were attacked by the military since July 8.
Israel has not provided details about targets, only saying it is hitting Hamas rocket launchers and command and control centers.
Among the dead in the strike on the Gaza City apartment tower Monday was Zakariya Abu Dagha, a leader of the Democratic Front for the Liberation of Palestine, one of several small PLO factions.
Six of the 11 killed were members of the same family, al-Kidra said. He said the family - a German-educated engineer, Ibrahim Kelani, his wife and four children - had sought refuge in the apartment tower after fleeing their apartment in a border area of Gaza that has come under heavy Israeli fire.
The Israeli military reported heavy fighting with Hamas in Gaza on Monday.
Three soldiers were killed in those clashes, while a female suicide bomber was shot before she could detonate her explosive belt among soldiers, the military said.
Another four Israeli soldiers were killed in a firefight when Hamas militants sneaked into Israel through a tunnel from Gaza.
The militants popped out of the ground close to an Israeli community near the border with Gaza before they were spotted by the military, the army said. Israeli media said 10 Hamas fighters were killed.
Since Israel's ground operation in Gaza began last week, Israeli soldiers have uncovered 45 shafts leading into about 16 underground tunnels, some as deep as 90 feet (30 meters), the military said.
Israel says the tunnels, some starting from homes and mosques, are a strategic threat because they reach inside Israel and demolishing them is a high priority
Gaza militants have fired more than 2,000 rockets at Israel in the past two weeks, including 130 on Monday, the army said.
Alhlou reported from Jerusalem. Associated Press Writer Ian Deitch in Jerusalem contributed to this report.