Israel's defence minister Yoav Gallant on Monday ordered a "complete siege" of the Hamas-ruled Gaza Strip, two days after shock attacks by the militant group killed more than 700 Israeli civilians.

Gallant is currently conducting an operational assessment at IDF (Israel Defence Forces) Southern Command, together with the Head of Southern Command Maj.-Gen. Yaron Finkelman.

"I have given an order - Gaza will be under complete siege. There will be no electricity, food or fuel [delivered to Gaza]. We are fighting barbaric [terrorists] and will respond accordingly," Gallant was quoted in a statement as saying.

Gaza depends on Israel largely for its basic supplies and such a decision will have far-reaching consequences for 2.3 million people living in the densely populated area.

While Hamas activists looked celebrating on the street of the coastal territory on Saturday, wary commoners were seen looking to stock up basic needs speculating about the imminent harsh attack.

The streets of Gaza for the last two days have looked completely deserted, much like most Israeli towns in the south.

The "complete siege" of the coastal Strip comes amid unabated air strikes being carried out by Israel in which some 500 Palestinians are said to have been killed.

The spokesperson for Hamas’s Izz ad-Din al-Qassam Brigades, going by the nom de guerre Abu Obeida, claimed that Israel’s bombing in the Gaza Strip has killed four Israeli hostages.

More than 100 Israelis, old, young and children have been taken captive by Hamas and Islamic Jihad in the attack which started on Saturday.

"The occupation’s bombing tonight and today on the Gaza Strip led to the killing of four enemy prisoners and the martyrdom of their captors, the Qassam Mujahideen,” Abu Obeida said in a statement on his Telegram channel.

Israel's Minister of Defence has been issuing stern warnings of a protracted fight since Saturday.

"Today we saw the face of evil. The Hamas [terrorist organization] has launched a brutal attack against the citizens of the State of Israel - attacking men, women, children and the elderly, indiscriminately. Hamas will understand very quickly that it has made a mistake - a grave mistake, and will pay a [heavy] price", he said soon after the brutal attacks.

"Fifteen years ago, as Head of the Southern Command, I came close to 'breaking the neck' of [destroying] Hamas. I was stopped by the political echelon. This phenomenon will not continue. We will change reality on the ground in Gaza for the next 50 years. What was before, will be no more. We will operate at full force", Gallant asserted.

After more than 50 hours of firefight, the Israeli army claimed that it has regained control of all Gaza border towns, but said that some militants who infiltrated may be remaining around.

Israel has imposed various levels of blockade on Gaza since Hamas seized power from rival Palestinian forces in 2007.

Israel intensifies Gaza strikes and scours south for Hamas fighters as death toll nears 1,200

Israel's military scoured the country's south for Hamas fighters and guarded breaches in its border fence with tanks on Monday, as it pounded the Gaza Strip from the air and mustered for a campaign its prime minister said would destroy “the military and governing capabilities” of the militant group.

More than two days after Hamas launched its unprecedented incursion from Gaza, the military said the fighting had largely died down for now.

The attack caught Israel's vaunted military and intelligence apparatus completely off guard, bringing heavy battles to its streets for the first time in decades.

Israel formally declared war on Sunday, portending greater fighting ahead, and a possible ground assault into Gaza — a move that in the past has brought intensified casualties. Palestinian militants continued firing barrages of rockets, setting off air raid sirens in Jerusalem and Tel Aviv.

Civilians are already paying a high price. Around 700 people have been killed in Israel — a staggering toll by the scale of its recent conflicts.

Nearly 500 have been killed in Gaza, a tiny, impoverished enclave of 2.3 million Palestinians bordering Israel and Egypt.

Palestinian militant groups claimed to be holding over 130 captives from the Israeli side.

Israeli Defence Minister Yoav Gallant meanwhile ordered a “complete siege” on Gaza, saying authorities would cut electricity and block the entry of food and fuel.

Israel and Egypt have imposed various levels of blockade on Gaza since Hamas seized power from rival Palestinian forces in 2007.

The chief military spokesman, Rear Adm. Daniel Hagari, told reporters Israel has “control” of its border communities.

He said there had been some isolated incidents early Monday, but that “at this stage, there is no fighting in the communities.” He added that “there might still be terrorists in the region.”

Israeli tanks and drones, meanwhile, guarded openings in the border fence to prevent more infiltrations, Hagari said, adding that 15 of 24 border communities have been evacuated, with the rest expected to be evacuated in over the next 24 hours.

Earlier, Hamas spokesman Abdel-Latif al-Qanoua told The Associated Press over the phone that the group's fighters continued to battle outside Gaza and had captured more Israelis as recently as Monday morning.

He said the group aims to free all Palestinian prisoners held by Israel, which in the past has agreed to painful, lopsided exchange deals in which it released large numbers of prisoners for individual captives or even the remains of soldiers.

Meanwhile, Israel hit more than 1,000 targets in Gaza, its military said, including airstrikes that levelled much of the town of Beit Hanoun in the enclave's northeast corner.

Hagari said Hamas was using the town as a staging ground for attacks. There was no immediate word on casualties, and most of the community's population of tens of thousands likely fled beforehand.

Hagari said the army had called up around 300,000 reservists — a massive mobilization — and that Israel would aim to end Hamas' rule of Gaza.

“Our task is to make sure that Hamas will no longer have any military capabilities to threaten Israel,” said spokesperson Jonathan Conricus in a video tweeted by Israel's military. “And in addition to that, we will make sure that Hamas is no longer able to govern the Gaza Strip.”

Hamas is deeply rooted in Gaza and has ruled the territory since driving out forces loyal to the internationally recognised Palestinian Authority in 2007.

Its rule has gone unchallenged through the 16-year Israeli and Egyptian blockade and four previous wars with Israel.

After breaking through Israeli barriers with explosives at daybreak Saturday, Hamas gunmen rampaged for hours, gunning down civilians and snatching people in towns, along highways and at a techno music festival attended by thousands in the desert.

Palestinian militants have also launched around 4,400 rockets at Israel, according to the military.

The Israeli military estimated 1,000 Hamas fighters took part in Saturday's initial incursion.

The high figure underscored the extent of planning by the militant group, which has said it launched the attack in response to mounting Palestinian suffering under Israel's occupation of the West Bank, its blockade of Gaza, its discriminatory policies in annexed east Jerusalem and tensions around a disputed Jerusalem holy site sacred to Muslims and Jews.

The Palestinians want a state of their own in all three territories, captured by Israel in the 1967 war, but the last serious peace talks broke down well over a decade ago, and Israel's far-right government is opposed to Palestinian statehood.

Hamas and the smaller Islamic Jihad group claim to have taken captive more than 130 people from inside Israel and brought them into Gaza.

The captives are known to include soldiers and civilians, including women, children and older adults, mostly Israelis but also some people of other nationalities. The Israeli military has said only that the number of captives is “significant.”

Mayyan Zin, a divorced mother of two, said she learned that her two daughters had been abducted when a relative sent her photos from a Telegram group showing them sitting on mattresses in captivity.

She then found online videos of a chilling scene in her ex-husband's home: Gunmen who had broken in speak to him near the two weeping daughters, Dafna, 15, and Ella, 8.

Another video showed the father being taken into Gaza.

“Just bring my daughters home and to their family. All the people,” Zin said.

The Israeli military was evacuating at least five towns close to Gaza, while the UN said more than 123,000 Gazans had been displaced by the fighting.

In Gaza, residents feared further escalation.

As of late Sunday, Israeli airstrikes had destroyed 159 housing units across the territory and severely damaged 1,210 others, the UN said.

The UN agency for Palestinian refugees, UNRWA, said a school sheltering more than 225 people took a direct hit. It did not say where the fire came from.

In the city of Rafah in southern Gaza, an Israeli airstrike early Monday killed 19 people, including women and children, said Talat Barhoum, a doctor at the local Al-Najjar Hospital.

Barhoum said aircraft hit the home of the Abu Hilal family, and that one of those killed was Rafaat Abu Hilal, a leader of a local armed group. The strike caused damage to surrounding homes.

Over the weekend, another airstrike on a home in Rafah killed 19 members of the Abu Quta family, including women and children, survivors said.

Several Israeli media outlets, citing rescue service officials, said those killed on the Israeli side include at least 73 soldiers.

The Gaza Health Ministry said 493 people, including 78 children and 41 women, were killed in the territory. Thousands have been wounded on both sides. An Israeli official said security forces have killed 400 militants and captured dozens more.

An Egyptian official said Israel sought help from Cairo to ensure the safety of the hostages.

Egypt also spoke with both sides about a potential cease-fire, but Israel was not open to a truce “at this stage,” according to the official, who insisted on not being identified because he was not authorized to brief media.

On Sunday, the US dispatched an aircraft carrier strike group to the Eastern Mediterranean to be ready to assist Israel, and said it would send additional military aid.

In northern Israel, a brief exchange of strikes with Lebanon's Hezbollah militant group fanned fears that the fighting could expand into a wider regional war. The Israeli military said the situation was calm after the exchange.

Elsewhere, six Palestinians were killed in clashes with Israeli soldiers Sunday around the West Bank.