"Hamas terrorists infiltrated Israel through two tunnels from north Gaza. IDF intercepted and killed more than 10 terrorists," Lieutenant Colonel Peter Lerner wrote on his official Twitter feed.
Israeli aircraft targeted one of the squads and soldiers opened fire on the other, resulted in the militants being killed.
Kerry makes mic gaffe on Gaza
Israelis soldiers flash the victory sign on their armoured personnel carrier (APC) at an army deployment near Israel's border with the Gaza Strip. (AFP photo)
UN, Obama want ceasefire as toll crosses 500
Meanwhile, the UN Security Council has called for an "immediate ceasefire" as Israel pressed on with a blistering assault on Gaza on Monday taking the Palestinian death toll above 502.
US President Barack Obama echoed the call in a telephone conversation with Israeli Prime Minister Benjamin Netanyahu early Monday, as Secretary of State John Kerry headed to the region to join true efforts.
A fresh air strike on Monday morning killed a family of nine, including seven children, after a weekend of incessant shelling by land, sea and air sent thousands of terrified civilians fleeing their homes.
A man was also killed in a strike on a motorbike, taking the Palestinian death toll to 502 as the Israeli offensive entered its 14th day, medics said.
Deadliest day yet of Israel-Gaza fighting
The Israeli army said 13 soldiers had been killed inside Gaza on Sunday, raising to 18 the number of soldiers killed since a ground operation began late on Thursday.
Gaza's emergency services spokesman Ashraf al-Qudra said many of the more than 150 Palestinians killed on Sunday -- the bloodiest day of fighting in Gaza in years -- were women and children.
The Security Council held urgent talks on the conflict, expressing "serious concern about the growing numbers of casualties."
"The members of the Security Council call for an immediate cessation of hostilities," said Rwandan ambassador Eugene Richard Gasana, whose country chairs the 15-member council.
Late Sunday, the armed wing of the Islamist Hamas movement, which dominates Gaza, claimed it had kidnapped an Israeli soldier, prompting celebrations in the streets of Gaza City and West Bank towns.
"The Israeli soldier Shaul Aaron is in the hands of the Qassam Brigades," a spokesman using the nom de guerre Abu Obeida said in a televised address.
Israel's UN ambassador Ron Prosor denied a soldier had been kidnapped, saying: "Those rumours are untrue."
However a spokeswoman for the Israeli military said they were investigating the claim.
Hamas militants held Israeli soldier Gilad Shalit for more than five years until his release in a 2011 prisoner exchange.
A medic helps a Palestinian in the Shejaia neighbourhood, which was heavily shelled by Israel during fighting, in Gaza City. (Reuters photo)
US Secretary of State John Kerry is making a renewed push for a cease-fire between Israel and Hamas with another trip to the Middle East after the civilian death toll in the conflict sharply escalated over the weekend.
Kerry left Washington early Monday for Cairo, where he will join diplomatic efforts to resume a truce that had been agreed to in November 2012.
The Obama administration, including Kerry, is sharpening its criticism of Hamas for its rocket attacks on Israel and other provocative acts, like tunneling under the border. It is also toning down an earlier rebuke of Israel for attacks on the Gaza Strip that have killed civilians, including children.
In a statement Sunday evening, State Department spokeswoman Jen Psaki said the US and international partners were "deeply concerned about the risk of further escalation, and the loss of more innocent life."
Two Americans, Max Steinberg of California and Nissim Carmeli of Texas, who fought for the Israel Defense Forces were killed in fighting in the Gaza Strip. The State Department confirmed the names of the two US citizens Sunday night.
Relatives of Palestinians killed during the overnight shelling of the eastern Gaza district of Shejaiya by the Israeli army arrive at the morgue of the al-Shifa hospital in Gaza City. (AFP photo)
During the Security Council talks, Palestinian envoy Ryad Mansour called for decisive steps to end the violence, and voiced frustration with what he termed the world body's failure to take a strong stand.
"The Council failed again and again to shoulder its responsibility," Mansour told reporters.
Prosor said Israel had agreed to earlier ceasefire proposals from Egypt and was acting in self-defence.
British Prime Minister David Cameron also spoke with Netanyahu, voicing "strong support for Israel's right to take proportionate action to defend itself from" Gaza rockets, while expressing condolences over the Israeli deaths and concern over the "mounting civilian casualties in Gaza."
Ceasefire discussions between Hamas leader Khaled Meshaal and Palestinian president in Qatar were due to take place on Monday after being delayed by a day.
Palestinian relatives grieve as mourners carry the bodies of two brothers who where killed at their family house in an airstrike, during their funerals in Rafah in the southern Gaza Strip. (AP photo)
'Too many innocents dying'
UN chief Ban Ki-moon was also in Doha where he urged Israel to "exercise maximum restraint".
"Too many innocent people are dying...(and) living in constant fear," he told a news conference in Doha.
So far, ceasefire proposals have been rejected by Hamas which has pressed on with its own attacks.
The Islamists have been demanding that as part of any truce there be a complete end to Israel's long-standing blockade od Gaza, the Rafah crossing with Egypt be opened, and Israel free scores of veteran prisoners re-arrested in recent weeks.
In the early hours of Monday Israeli jets carried out strikes throughout Gaza, including in southern Rafah and Khan Yunis, and northern Beit Lahiya.
Witnesses said warplanes had bombed the home of Hamas politburo member Imad al-Alami in Gaza City, as well as the Khan Yunis home of head of security and police in Gaza, General Saleh Abu Sharekh. No casualties were reported in either attack.
On Sunday, thousands of Palestinians fled the Shejaiya neighbourhood, between Gaza City and the Israeli border, after hours of heavy shelling which killed 72 people, an AFP correspondent reported.
The UN agency for Palestinian refugees UNRWA has opened 61 of its schools to shelter more than 81,000 people who have fled their homes.
The Doctors Without Borders charity urged Israel to "stop bombing civilians trapped in the Gaza Strip", noting the majority of the injured arriving in the Al-Shifa hospital were women and children.
A picture taken from the Israeli Gaza border shows an Israeli army flare illuminating the sky above the Gaza strip. (AFP Photo)
Netanyahu blames Hamas
Netanyahu has blamed the civilian casualties on Hamas using innocents "as human shields."
He insisted on Sunday that the military campaign had strong international backing.
"We are carrying out a complex, deep, intensive activity inside the Gaza Strip and there is world support for this... very strong support," he said before a security cabinet meeting.
Although Israel said earlier Sunday it was expanding its ground operation to destroy tunnels used by militants to carry out cross-border attacks and fire rockets, Netanyahu said troops could end their mission "fairly quickly".
His Defence Minister Moshe Yaalon also suggested it could end within days, demanding international action to "demilitarise Gaza".
(REUTERS and AFP inputs)