NEAR THE ISRAEL-GAZA BORDER (CNN) - People on both sides of the raging violence between Israel and Hamas were given hope Tuesday that a halt might be imminent. But Egypt and Hamas -- which had announced the potential "calming" -- soon seemed to pull back on their statements.
Meanwhile, Israel suffered its first military fatality in the week of violence, another child was among the dead in Gaza, and CNN saw a group of men in Gaza drag the body of a man through streets from the back of a motorcycle. The men -- who carried weapons -- yelled in Arabic that he was a traitor and Israeli spy.
Israel said it was holding off on a ground offensive into Gaza to give diplomatic efforts time. Those efforts include talks that U.S. Secretary of State Hillary Clinton and U.N. Secretary-General Ban Ki-moon each planned with Israeli officials and with Palestinian officials in the West Bank, where the government is run by the Palestinian faction Fatah. Gaza is run by Hamas, which the United States and numerous other countries consider a terrorist organization.
In Jerusalem, Clinton and Israeli Prime Minister Benjamin Netanyahu spoke to reporters.
"President Obama asked me to come to Israel with a very clear message," Clinton said. "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."
Netanyahu thanked the secretary for her support, and accused "terrorist enemies" of "doing everything in their power to maximize the number of civilian casualties."
"Obviously, no country can tolerate a wanton attack on its civilians," he said. "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."
Hamas spokesman Sami Abu Zuhri announced that Israel has not agreed to terms that would stop the fighting. Earlier, a senior Hamas official told CNN a "calming down" would be announced at 9 p.m. local time (2 p.m. ET). That did not happen.
A "calming down" could halt violence, but is not the same as an official cease-fire or truce, CNN's Ben Wedeman reported.
Egyptian President Mohamed Morsy said earlier Tuesday that 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.
Egypt has been working to broker a cease-fire that would end seven days of deadly violence that has turned life into a nightmare for millions in Israel and Gaza.
Israel has said it wants a cease-fire agreement, but has not indicated whether one could be imminent.
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 hasn't arrived.
Regev said Israel is not interested in a "time out," allowing Hamas to regroup after Israeli strikes have done damage. "We want a new reality" in which Israelis don't live under rocket fire from Hamas, he said.
To succeed, negotiations have to be done "discreetly," he said.
Hamdan said Hamas' actions have been "a good lesson for the Israeli government. It's not good to attack the Palestinians, expecting that they will not react against the attack."
Asked whether Hamas would accept Israel's right to exist, Hamdan said the Palestinian people would not consider it without an end to occupation.
Earlier in the day, Mohammed al-Deif, a commander of the al Qassam Brigades -- Hamas' military wing -- said of Israel, "The ground operation that they keep threatening of waging will be the greatest hope to release our prisoners." He said the brigades' operation "will be the starting point for the next phase to liberate Palestine."
On Twitter, the brigades quoted him as saying the "Zionist enemy would pay (a) high price for his crimes in Gaza."
The brigades tweeted that it was shelling numerous areas in Israel.
The onslaught of rockets fired Tuesday into Israel continued to be met by Israeli strikes on sites in Gaza. Attacks Tuesday included one aimed at Jerusalem, one that caused casualties in the southern town of Beer Sheva, and one that injured five Israeli soldiers. Another rocket hit a civilian building in Rishon LeZion, part of metropolitan Tel Aviv, IDF spokeswoman Avital Leibovich said. The building was partly set on fire, and one person was probably lightly wounded, she said.
Israel's Netanyahu, at a news conference with Ban, said Israel was fighting back with "surgical
operations against terrorists at a time when our own population is being bombarded by rocket attacks."
"If we hope to make these tactics illegitimate, they should be condemned in the most forceful terms by all responsible members of the international community," he said. "The moment we draw symmetry between the victims of terror and the unintended casualties that result from legitimate military action against the terrorists, the minute that false symmetry is drawn, the terrorists win."
Ban said he visited Israeli cities struck by rocket fire and met with people "living in fear and terror."
"This is unacceptable," he said.
While noting Netanyahu's statement that Israel's military works to avoid civilian casualties, Ban added, "The loss of civilian lives is unacceptable under any circumstances.
"The excessive use of force is unlawful and must be rejected."
Jerusalem Mayor Nir Barkat said Ban was among those who had to take cover when sirens sounded in Jerusalem, warning of an incoming rocket. Military officials said the rocket landed in an open area of a village.
The Gaza Ministry of Health said 27 people were killed Tuesday, including children. The ministry said 137 Palestinians have been killed and more than 1,100 injured since the latest hostilities erupted seven days ago.
Officials have not said how many militants were killed.
According to Israel's Soroka Hospital, five people have been killed in Israel. More than 70 have been injured, Israeli officials said, and soldiers were among the wounded.
The death toll moved up from three to four early Tuesday when an 18-year-old soldier was killed in a rocket attack in the town of Eshkol, the IDF said. It climbed from four to five, according to the hospital, when a Bedouin of unknown age was killed.
Other violence Tuesday included 11 Hamas rockets that hit the Israeli city of Beer Sheva, causing casualties. More than 30 rockets were fired into the area Tuesday, but most were destroyed by Israel's Iron Dome interceptors.
CNN's Wolf Blitzer spoke with a father, mother and four children who had heard the warning siren Israel uses to let people know a rocket is incoming. They ran to a safe room and survived by only a matter of about 30 seconds.
In Tel Aviv, a man with an ax attacked a U.S. Embassy security guard, Israeli police said. The attacker, who also had a knife, was arrested, police spokesman Micky Rosenfeld said.
Meanwhile, a strike by Israel "targeted two terrorists in the northern Gaza Strip," and a "direct hit was confirmed," the IDF said. Israeli military forces also "targeted a terrorist squad in Gaza that was preparing to fire rockets at Israel," the IDF said.
The IDF again dropped leaflets across parts of Gaza warning residents to evacuate their homes and go to central Gaza City. The leaflets told them which way to go and what part of the city to stay in once they arrived.
The leaflets are part of Israel's efforts to minimize civilian casualties, Israeli officials say.
As a CNN crew was getting video of families fleeing an area in Gaza, an explosion shook the windows of a school, CNN's Arwa Damon reported. The target was a vehicle around the corner, and the strike killed two people, she said.
Throughout Tuesday, a fresh barrage of airstrikes pummeled Gaza.
The Israel Defense Forces said it targeted 100 sites overnight, "including underground rocket launchers, terror tunnels and ammunition storage facilities." The IDF said the "terror tunnels" were used as hideouts by Hamas operatives.
Israel has sent at least 1,350 airstrikes to Gaza, the IDF said.
"The mere fact that Hamas shoots a rocket at Jerusalem -- it says more about them that it does about anything else," Regev, the Israeli spokesman, said Tuesday after the sirens had stopped sounding in the city. He pointed out that Jerusalem has Jewish neighborhoods, Arab neighborhoods and other communities.
Iran is arming Hamas with rockets, he said, echoing remarks by Israeli President Shimon Peres on CNN.
Gaza has endured a crippling economic embargo since Hamas won control of the territory from the Fatah-led Palestinian Authority after a landslide 2007 election that was followed by intra-Palestinian clashes.
Many Arab and Muslim nations view Hamas as the victim of Israeli aggression.
Diplomats hope to avoid a repeat of 2008, when at least 1,400 people died when Israeli troops invaded Gaza after a similar spate of rocket attacks.