WASHINGTON >> Secretary of State John Kerry is heading back to the Middle East as the Obama administration attempts to bolster regional efforts to reach a ceasefire and sharpens its criticism of Hamas in its conflict with Israel.

The State Department said Kerry would leave early Monday for Egypt where he will join diplomatic efforts to resume a truce that had been agreed to in November 2012. In a statement Sunday evening, department spokeswoman Jen Psaki called the U.S. and international partners “deeply concerned about the risk of further escalation, and the loss of more innocent life.”

The Obama administration has toned down its earlier rebuke of Israel for attacks on the Gaza Strip that have killed civilians, including children, although both President Barack Obama and Kerry expressed concern about the rising death toll.

The U.S. will urge the militant Palestinian group to accept a cease-fire agreement that would halt nearly two weeks of fighting with Israel. More than 430 Palestinians and 20 Israelis have been killed in that time.

Cairo has offered a cease-fire plan that is backed by the U.S. and Israel. But Hamas has rejected the Egyptian plan and is relying on governments in Qatar and Turkey for an alternative proposal. Qatar and Turkey have ties to the Muslim Brotherhood, which is also linked to Hamas but banned in Egypt.

Making the rounds of Sunday talk shows, Kerry pointed to Hamas’ role in the violence.

“It’s ugly. War is ugly, and bad things are going to happen,” Kerry told ABC’s “This Week.” But, he added, Hamas needs “to recognize their own responsibility.”

Both Obama and Kerry said Israel has a right to defend itself against frequent rocket attacks by Hamas from the Gaza Strip. Kerry accused Hamas of attempting to sedate and kidnap Israelis through a network of tunnels that militants have used to stage cross-border raids.

He said on CNN’s “State of the Union” that Hamas must “step up and show a level of reasonableness, and they need to accept the offer of a cease-fire.”

Then, Kerry said, “we will certainly discuss all of the issues relevant to the underlying crisis.”

The nearly two-week conflict appeared to be escalating as U.N. chief Ban Ki-moon was already in the region to try to revive cease-fire efforts.

Obama, in a telephone call Sunday, told Israeli Prime Minister Benjamin Netanyahu that Kerry was coming to the Mideast and condemned Hamas’ attacks, according to a White House statement.

The U.N. relief agency in Gaza estimates that 70,000 Palestinians have fled their homes in the fighting and are seeking shelter in schools and other shelters the United Nations has set up. The relief agency’s top director in Gaza, Robert Turner, told CNN’s “State of the Union” that the U.N. has run out of mattresses for refugees and few hygiene and medical supplies are left, although fresh food and water remain available.

“People are scared,” Turner said. “They don’t feel safe at home, they don’t feel safe with their families or neighbors. They feel relatively safe in our installations. … We frankly have been overwhelmed by the numbers.”

He said more than 1,000 homes in Gaza have been destroyed or damaged beyond repair, and at least 13,000 lightly damaged.

U.S. officials made clear, however, that Hamas could bring relief to the Palestinian people if it agrees to a cease-fire proposed by Egypt — a view that Netanyahu is pushing as well.

Republican Sen. Lindsey Graham, a member of the Senate Armed Forces Committee, urged Israel to “stay as long as you need to stay, go wherever you need to go, do deal with a viper’s nest called Hamas.”

“If it’s left up to Hamas, thousands of Israelis would be dead,” Graham, R-S.C., told NBC’s “Meet the Press.”

Netanyahu agreed. In an ABC interview, he said Israel has tried to avoid killing Palestinian civilians through phone calls, text messages and leaflets dropped on their communities.

But Hamas doesn’t “give a whit about the Palestinians,” Netanyahu said. “All they want is more and more civilian deaths.”

The prime minister said his top goal is to restore a sustainable peace, but then will ask the international community to consider demilitarizing Gaza to rid Hamas of its rockets and shut down the tunnels leading into Israel. Netanyahu brushed off a question about giving concessions to Hamas as a step toward peace, including releasing Palestinian prisoners or loosening border crossings.

“Hamas doesn’t care,” Netanyahu said. “I think the last thing you want to do is reward them.”

Kerry also said any cease-fire agreement must be without conditions or “any rewards for terrorist behavior.” He did not mention the Qatari or Turkish efforts.

Kerry also blamed the latest wave of violence on what he called Israel’s “legitimate” efforts to pursue and punish those who last month kidnapped and killed three Israeli teenagers whose bodies were found in the West Bank.

Their deaths were followed almost immediately by what authorities believe was a retribution attack on a Palestinian youth who was strangled, beaten and burned to death.

Tensions between Israel and Palestinian authorities have been simmering for years. They threatened to boil over this spring when Israel shelved nearly nine months of peace negotiations that were being personally shepherded by Kerry after Palestinian President Mahmoud Abbas agreed to create a unity government with Hamas.

“No country could sit by and not take steps to try to deal with people who are sending thousands of rockets your way,” Kerry said.

Kerry spoke Sunday on all five major news network talk shows: NBC’s “Meet the Press,” CNN’s “State of the Union,” ABC’s “This Week”, CBS’ “Face the Nation” and “Fox News Sunday.”