Hopes were briefly raised then dashed Tuesday for an imminent calm between Israel and Hamas as diplomats fervently sought a way to stop the deadly violence.

U.S. Secretary of State Hillary Clinton arrived in the region, where she met for more than two hours with Israeli Prime Minister Benjamin Netanyahu. She is expected to meet later with Palestinian Authority President Mahmoud Abbas and Egyptian President Mohamed Morsy, who is working to broker a cease-fire.

Gaza is run by Hamas, which the United States and other countries consider a terrorist organization.

Clinton offered Israel the support of the United States and expressed hope for a lasting solution as she spoke to reporters alongside Netanyahu at the prime minister's office in Jerusalem. She thanked Egypt and especially Morsy for their efforts.

"President Obama asked me to come to Israel with a very clear message. America's commitment to Israel's security is rock solid and unwavering. That is why we believe it is essential to de-escalate the situation in Gaza," she said. "The goal must be a durable outcome that promotes regional stability and advances the security and legitimate aspirations of Israelis and Palestinians alike."

Tuesday marked the seventh day of deadly violence that that has turned life into a nightmare for millions in Israel and Gaza. Rockets rained as the death toll continued to climb.

There was a moment when it looked like the attacks might stop. A senior Hamas official told CNN a "calming down" would be announced at 9 p.m. local time (2 p.m. ET). But that did not happen.

Hamas spokesman Sami Abu Zuhri later said that Israel had not agreed to terms that would stop the fighting.

Similarly, Egypt's Morsy said the "travesty of the Israel aggression on Gaza will end in a few hours." But a few hours passed, and Morsy's office told CNN not to expect any announcement Tuesday night.

A "calming down" could halt violence, but it is not the same as an official cease-fire or truce. Israel has said it wants a cease-fire agreement but has not indicated whether one could be imminent.

"Obviously, no country can tolerate a wanton attack on its civilians," said Netanyahu, standing beside Clinton. "Now, if there is a possibility of achieving a long-term solution to this problem through diplomatic means, we prefer that. But if not, I'm sure you understand that Israel will have to take whatever actions necessary to defend its people."

Netanyahu accused "terrorist enemies" of "doing everything in their power to maximize the number of civilian casualties."

According to the Gaza Ministry of Health, 27 people were killed Tuesday, including children, and 137 Palestinians have been killed since the latest hostilities erupted. More than 1,100 people were injured. It's not clear how many of the victims were militants.

Five people have been killed in Israel, according to Israel's Soroka Hospital. More than 70 have been injured, including soldiers, Israeli officials have said.

The death toll in Israel moved from three to four early Tuesday when an 18-year-old soldier was killed in a rocket attack in the town of Eshkol, officials said. It climbed from four to five, according to the hospital, when a 30-year-old Bedouin was killed.

Israel said it was holding off on a ground offensive into Gaza to give diplomatic efforts time. Those efforts include talks with Clinton and U.N. Secretary-General Ban Ki-moon.

A source familiar with discussions in Jerusalem told CNN, "One of the Israeli demands is that there should be a period of total calm for 24 hours before committing to any agreement."

Israeli spokesman Mark Regev told CNN, "Diplomacy is still ongoing."

"It's in the hands now of the Israelis," Osama Hamdan, Hamas spokesman in Beirut, told CNN earlier by phone. "... I think the Egyptians are waiting for some support, promised support, from the United States in order to make an end for that. So we expect to have an outcome of this issue today as President Morsy has said." That "expected" outcome didn't arrive.

Speaking early Wednesday, Hamdan said he believes Hamas and Israel are close to a cease-fire agreement.