United Nations Secretary-General Ban Ki-moon is expressing alarm at the continuing violence between Israelis and Palestinians and demanding both sides take immediate steps to end the fighting.
In a statement late Sunday, Ban said rocket fire by Hamas militants into Israel goes against international law, while "too many Palestinians have been killed" by Israeli airstrikes. He said any potential Israeli ground offensive would surely lead to more casualties.
Palestinian officials say more than 160 people have been killed, including militants, since Israel launched its offensive to end Hamas rocket fire last Tuesday.
The U.N.'s children's agency said Monday that 33 children are among those who have died, and that young people are "bearing the brunt" of the violence.
It joined the U.N. Security Council's call from Saturday for an immediate cease-fire.
Thousands fled their homes in a Gaza town on Sunday after Israel warned them to leave ahead of threatened attacks on rocket-launching sites, on the sixth day of an offensive that Palestinian officials said has killed at least 160 people.
"Those who fail to comply with the instructions will endanger their lives and the lives of their families. Beware," read a leaflet dropped by the Israeli military in the town of Beit Lahiya, near the border with Israel.
Despite intensified Israeli military action - which included a commando raid overnight in what was Israel's first reported ground action in Gaza during the current fighting - militants continued to launch rocket after rocket across the border.
Commandos targeted what Israel called a rocket-launching site. Authorities said four Israeli soldiers were slightly wounded during an exchange of gunfire with Palestinian fighters.
Palestinians take shelter at the United Nations school after they fled their homes in Gaza City, July 13, 2014.
Israel Prime Minister Benjamin Netanyahu, appearing on CBS' Face the Nation on Sunday, discussed Israel's goals for the offensive, but not the intended methods to achieve them.
“Our goal is to achieve sustainable quiet and security for our people, and we are doing that by degrading Hamas and other terror capabilities. I am not going to say right now how or when that goal will be achieved," Netanyahu said.
Neither Israel nor Palestinian militants show signs of agreeing to a cease-fire, despite calls by the United Nations Security Council and others to end the increasingly bloody offensive that has claimed at least 160 Palestinian lives and injuring about 1,000.
Several Israeli citizens have been wounded, but no one killed, by Palestinian rockets, which are being launched at longer range than in previous conflicts. Israeli defense forces credit their Iron Dome missile defense system, which intercepts rockets in mid-air, with preventing greater casualties.
With Israel massing tanks and soldiers at Gaza's borders, some fear that could signal a wider ground offensive that would cause heavy casualties.
Drone shot down
Israel said it shot down a drone from Gaza on Monday, the first reported deployment of an unmanned aircraft by Palestinian militants whose rocket attacks have been regularly intercepted.
Hamas said its armed wing had sent several drones to carry out “special missions” deep inside Israel - a development which, if confirmed, would mark a step up in the sophistication of its arsenal.
More than 166 Palestinians, most of them civilians, have been killed, Gaza health officials said, in seven days of fighting that has shown no sign of ending.
The Israeli military said the drone was intercepted near the port of Ashdod by a U.S.-built Patriot missile, used largely ineffectively by Israel against Iraqi Scud missiles in the 1991 Gulf War.
The force was trying to locate debris in the area about 15 miles north of Gaza, and determine whether it had carried explosives.
Hundreds of airstrikes
Israel has launched more than 1,300 airstrikes since the offensive began, military spokesman Lt. Col. Peter Lerner said Sunday.
Palestinian militants have launched more than 800 rockets at Israel, including 130 in the last 24 hours, the Israeli military said Sunday.
Israeli leaflets dropped on Beit Lahiya, where 70,000 Palestinians live, said civilians in three of its 10 neighborhoods were "requested to evacuate their residences" and move south, deeper into the Gaza Strip, by 12 p.m. (0900 GMT).
The Gaza Interior Ministry, in a statement on Hamas radio, dismissed the Israeli warnings as “psychological warfare” and instructed those who left their homes to return and others to stay put.
The warnings cited roads that residents could use safely and said Israeli forces intended to attack "every area from where rockets are being launched."
It was not clear whether the possible attack would be confined to stepped-up airstrikes or whether it might include a sizeable ground offensive - something that Israel has so far been reluctant to undertake.
It was the first time Israel had warned Palestinians to vacate dwellings in such a wide area.
Previous warnings, by telephone or so-called "knock-on-the-door" missiles without explosive warheads, had been directed at individual homes slated for attack.
At least 4,000 people fled Beit Lahiya and crowded into eight U.N.-run schools in Gaza City on Sunday, a spokesman for the United Nations Relief and Works Agency said.
At schools run by the United Nations Relief and Works Agency in Gaza City, Beit Lahiya residents arrived in donkey carts filled with children, luggage and mattresses, while others came by car or taxi. One man, still in his pajamas, said some inhabitants had received phone calls warning them to clear out.
"What could we do? We had to run in order to save the lives of our children," said Salem Abu Halima, 25, a father of two.
Many foreign nationals and Palestinians with duel nationalities are leaving the Gaza Strip.
An Israeli police explosives expert carries the remains of a rocket that landed in Ashkelon, Israel, July 13, 2014.
Israel also continued its airstrikes, targeting areas across Gaza early Sunday, ignoring international appeals for a cease-fire as the conflict entered its sixth day.
“We don't know when the operation will end,” Netanyahu told a Cabinet meeting Sunday. “It might take a long time.”
A Palestinian woman and a girl aged 3 were killed in Israeli air strikes early on Sunday, the Gaza Health Ministry said.
Overnight, an Israel strike overnight flattened the home of the Gaza police chief, killing at least 18 people in the deadliest attack since Israel launched its offensive on Gaza Tuesday.
The strike also badly damaged a nearby mosque, wounding scores of people. Some worshippers are still believed to be trapped in the rubble.
An earlier airstrike Saturday also hit a mosque. Israeli Defense Forces spokesperson Lieutenant Libby Weiss said the building was being used by Hamas militants.
"It's important to note that this was a mosque that was used almost entirely for Hamas operations. Meaning as a structure perhaps, yes it was a mosque, but it was very clear that it was being used, and I would say almost hijacked by Hamas for terror purposes," Weiss said.
The United Nations humanitarian affairs office has said 135 of the 160 Palestinians killed in Israeli strikes since operations began have been civilians.
Netanyahu said Friday Israel is considering all military options despite international calls for a cease-fire.
With the worsening situation in Gaza, Palestinian President Mahmoud Abbas said he has appealed to United Nations Secretary-General Ban Ki-Moon for "international protection" for the Palestinian people.
On Sunday, Netanyahu blamed Hamas for the ongoing military offensive in the Gaza Strip, saying responsibility for any casualties lies with the Palestinian militants.
"Who is hiding in mosques? Hamas. Who puts weapons arsenals beneath hospitals? Hamas. Who puts headquarters in residential areas or near primary schools? Hamas," Netanyahu said.
"Hamas is using the residents of Gaza as human shields and is bringing a catastrophe upon the citizens of Gaza and that's why the responsibility for any casualties among the civilians of Gaza, which Israel regrets, that responsibility lies with Hamas and its collaborators," he said.
Netanyahu went on to blame Iran for funding, arming and training Hamas, saying that the meeting in Vienna on Iran's nuclear program must prevent the Iranians from developing a nuclear weapon.
Palestinians flee their homes to take shelter at the United Nations school in Gaza City, July 13, 2014.
If they fail to do so, Netanyahu said, "What is happening around here and what is happening in the Middle East, will be several times worse or maybe beyond that."
Britain, France and Turkey have expressed concerns about the loss of life in Gaza. U.S. President Barack Obama said the U.S. is willing to help facilitate a cessation of hostilities, while backing "Israel's right to defend itself.
Dr. Mahdi Abdul Hadi, head of a Palestinian academic society in Jerusalem (PASSIA), said both Israeli and Palestinian leadership have shown little willingness to back down.
"Netanyahu is refusing any reconciliation, any negotiations, any halting of the operations and he is carrying on the mission as usual," Hadi said. "The same thing, Hamas is going forward as planned, saying whatever will happen will happen to all of us."
Israel claims to have hit more than 1,100 targets inside Gaza since operations began. The military has called up more than 30,000 reservists and is now weighing whether to begin ground operations in Gaza.
German Foreign Minister Frank-Walter Steinmeier will travel to the Middle East on Monday, a foreign ministry spokeswoman said, where he will meet Netanyahu and Palestinian President Mahmoud Abbas, German media reported.
Steinmeier called for an end to “Hamas rocket terror,” in comments to newspaper Bild am Sonntag published on Sunday, and urged a “coalition of reason” to stop the conflict escalating.
UN calls for cease-fire
The U.N. Security Council called for a cease-fire in the Israeli-Palestinian conflict.
In a statement Saturday, the 15-member council called for a de-escalation of the violence, a restoration of calm, and a reinstitution of a 2012 Egyptian-brokered cease-fire between the two sides.
This is the first official response to the crisis by the powerful U.N. body, which has been divided on the Israeli-Palestinian conflict.
Western foreign ministers meeting on Sunday said a ceasefire was an urgent priority.
Also Sunday, Pope Francis led tens of thousands of people in St. Peter's Square in unannounced, silent prayer for an end to the warfare in the Middle East, the Associated Press reported.
Francis called for insistent prayers for peace in the Holy Land during his Sunday window appearance to pilgrims and tourists.
He said he didn't consider "in vain" the June 8 peace prayer gathering he hosted at the Vatican attended by the Israeli and Palestinian presidents.
Francis, who made a Middle East pilgrimage in May, urged people to avoid concluding that "violence and hatred win out over dialogue and reconciliation."
Hostilities along the Israel-Gaza frontier first intensified last month after Israeli forces arrested hundreds of Hamas activists in the Israeli-occupied West Bank following the abduction there of three Jewish teenagers who were later found killed.
A Palestinian youth was then killed in Jerusalem in a suspected revenge attack by Israelis.
WATCH: Gaza Fighting Takes Civilian Toll - report by Zlatica Hoke
Some information for this report comes from AP, AFP and Reuters.