Kuwaiti Ambassador Mansour Al-Otaibi told reporters after the closed consultations late Tuesday that "we were not able to find a solution to the situation in Gaza" because of divisions in the most powerful U.N. body.
Al-Otaibi said that "the majority of member states raised the issue that the council should do something, and one of the issues that has been introduced is that the Security Council should consider going on a visit to the occupied territories."
Council diplomats said Swedish Ambassador Olof Skoog raised the idea of a trip, which was endorsed by Russia and Britain. The diplomats spoke on condition of anonymity because the discussions were private.
Kuwait and Bolivia called for the closed Security Council consultations.
Israel's U.N. ambassador says Security Council members that want to blame Israel for the latest fighting with the Hamas militant group in Gaza are "morally bankrupt" and rejects those who call for restraint by both sides.
Danny Danon told reporters at U.N. headquarters in New York that "there is no such thing as both sides."
He added: "There is Hamas that attacks and fires over 460 missiles at civilians and there is Israel that protects its people."
Danon said ahead of closed Security Council consultations late Tuesday that the council must condemn only Hamas "for its aggressive assault on civilians."
Asked about an undercover Israeli military operation that sparked the Hamas attacks and Israeli bombing of Gaza, Danon said: "We take action to protect our people and we will continue to do that."
The Palestinian U.N. ambassador says Egyptian mediators are in Israel trying to get the government to agree on a Gaza cease-fire.
Riyad Mansour, who represents the Western-backed Palestinian President Mahmoud Abbas, told reporters at U.N. headquarters in New York on Tuesday that "we are grateful for the Egyptians who are mediating."
The Islamic militant group Hamas, which rules Gaza, says it agreed to the Egyptian cease-fire proposal. There has been no word yet from Israel.
Mansour says "the Palestinian side is holding the quietness in order to allow for the Egyptian efforts to succeed."
He says Abbas is returning from Kuwait for emergency meetings.
Mansour condemned Israel for starting the cycle of violence with a botched undercover military operation on Sunday.
At the Palestinians' request, he said, Kuwait and Bolivia have asked for a meeting of the U.N. Security Council on the Israeli actions. China, the current council president, has called for closed consultations late Tuesday.
Some Palestinians in Gaza are celebrating and Israelis in southern Israel are protesting an emerging cease-fire deal to end more than 24 hours of heavy fighting.
Gazans see the cease-fire as a win for Hamas, for having brought Israel to heel with a barrage of rocket fire. Amid an apparent lull in violence, hundreds of Hamas supporters took to the streets in sporadic gatherings, cheering for the group's military wing.
In the southern Israeli town of Sderot, which has been pounded by rockets, dozens of protesters burnt tires, blocked traffic and chanted "disgrace" at what they saw as the government's caving in to militant violence.
While there has been no word from Israel, Gaza militants earlier Tuesday announced they had accepted an Egyptian-brokered cease-fire.
Palestinian militant groups in the Gaza Strip say they have accepted an Egyptian-mediated cease-fire amid a major flare-up of violence with Israel.
Hamas and smaller armed groups announced the cease-fire in a joint statement late Tuesday.
The deal would end the heaviest round of fighting between Israel and Hamas since a 2014 war.
Terms of the deal were not immediately known, and there was no immediate comment from Israel.
The leader of the Hamas militant group has signaled a readiness to halt the latest round of fighting with Israel.
Ismail Haniyeh said Tuesday that his group will stop its rocket fire on Israel if Israel halts its airstrikes.
In a statement, Haniyeh said "the Palestinian resistance is defending its people and itself before the Israeli aggression," adding that "if the occupation stops its aggression, it's possible to return to the cease-fire understandings."
Gaza militants have fired hundreds of rockets into Israel over the past 24 hours as Israel has struck militant targets across Hamas-ruled Gaza. The violence was triggered by a botched Israeli military raid on Sunday in which seven Palestinians and an Israeli officer were killed.
Israel's Security Cabinet has ordered the military to continue strikes in Gaza "as is necessary," signaling that the latest flare-up of violence may be far from over.
Israeli Prime Minister Benjamin Netanyahu convened his Security Cabinet Tuesday to plot a way forward amid the fighting. Rocket fire and airstrikes continued throughout the six-hour meeting.
The Israeli military says some 400 rockets and mortar shells have been launched from Gaza since Monday afternoon, with about 100 of them intercepted by Israel's Iron Dome defense system.
Israel said it has struck more than 100 targets it says are linked to militants in Gaza, including a strike that destroyed the TV station of the ruling Hamas militant group.
At least seven Palestinians, among them five militants, have been killed during the fighting. A Palestinian man working in Israel was killed after a rocket hit the building he was in.
Israeli media are reporting that a meeting of the Security Cabinet has ended after more than six hours.
Israeli Prime Minister Benjamin Netanyahu convened the meeting of ministers Tuesday to plot a way forward amid an intense flare-up of fighting with the Islamic militant group Hamas, which rules the Gaza Strip.
No official statement was released following the meeting, and it remained unclear what if anything had been decided.
The heavy exchange of fire, which was triggered by a botched Israeli undercover raid into Hamas-ruled Gaza late Sunday, marked the most serious escalation since an Israel-Hamas war in 2014.
The man killed in a Palestinian rocket attack in southern Israel has been identified as a 48-year-old Palestinian laborer who had been working in the Israeli city of Ashkelon.
Relatives say Mahmoud Abu Asbeh had worked in Ashkelon for 15 years, leaving his West Bank village of Halhoul each Sunday and returning home on weekends.
His cousin Jihad Abu Asbeh called it a "big shock" and says the entire village is sad. He says it was "God's will and there's nothing we can do about it."
Mahmoud's body was found in the rubble of a building struck by a rocket fired from Gaza. His cousin says Mahmoud was married with six children.
The armed wing of Hamas says it will step up attacks and fire rockets at Israeli towns of Ashdod and Beersheba if Israel continues to carry out airstrikes in Gaza.
The warning was issued on Tuesday, hours after a rocket fired from Gaza killed an Israeli man in a direct hit on a residential building in the coastal town of Ashkelon. The rocket was fired after an Israeli airstrike destroyed Hamas' TV building in Gaza.
The spokesman for the Hamas military wing, identified only as Abu Obeida, said in a statement that Ashkelon "has entered the range of fire as a response to the bombing of buildings in Gaza."
He says that Ashdod and Beersheba "are the next targets if the enemy continues bombing civilian buildings."
Of the three cities, Ashkelon is closest to Gaza.
Israeli medical services say a man has been killed by a rocket that hit an apartment building in southern Israel and a woman was critically wounded.
Magen David Adom says the two were trapped in an apartment in Ashkelon that was struck by a rocket from Gaza early on Tuesday. It's the first casualty in Israel from the most intensive wave of attacks since the 2014 Gaza war.
Gaza militants pummeled Israel overnight with dozens of rockets and the Israeli military responded with strikes against militant targets in Gaza.
The military says there were some 370 launches from Gaza since the current round began on Monday afternoon. About 100 of them were intercepted by Israel's Iron Dome defense system.
Israel has struck more than 100 militant targets in Gaza.