Hamas, the Islamist movement controlling Gaza, and Egypt, whose new, Islamist government is trying to broker a truce, had floated hopes for a ceasefire by late on Tuesday; but by the time Clinton met Israeli Prime Minister Benjamin Netanyahu it was clear there would be more argument, and more violence, first.
Hamas leaders in Cairo accused the Jewish state of failing to respond to proposals and said an announcement on holding fire would not come before daylight on Wednesday. Israel Radio quoted an Israeli official saying a truce was held up due to "a last-minute delay in the understandings between Hamas and Israel".
An initial halt to attacks may, however, not see the sides stand their forces down from battle stations immediately; Clinton, who flies to Cairo to see Egyptian President Mohamed Mursi later on Wednesday, spoke of a deal "in the days ahead".
As she arrived in Israel after nightfall, Israel was stepping up its bombardment. Artillery shells and missiles fired from naval gunboats offshore slammed into the territory and air strikes came at a frequency of about one every 10 minutes.
After seven days of hostilities that have killed over 130 Palestinians and five Israelis, two of these on Tuesday, both sides are looking for more than a return to the sporadic calm that has prevailed across the blockaded enclave since Israel ended a much bloodier air and ground offensive four years ago.
Netanyahu, who faces an election in two months that he is, for now, favoured to win, told Clinton he wanted a "long-term" solution. Failing that, Netanyahu made clear, he stood ready to step up the military campaign to silence Hamas's rockets.
Hamas for its part is exploring the opportunities that last year's Arab Spring has given it to enjoy favour from the new Islamist governments of states once ruled by US proteges, and from Sunni Gulf powers keen to woo it away from Shi'ite Iran. It has used longer-range missiles, some sent by Tehran, and hopes to eclipse Western-backed Palestinian President Mahmoud Abbas.
Hamas has spoken of an easing of Israel's blockade on the 40-km (25-mile) slice of Mediterranean coast that is home to 1.7 million people. It may count on some sympathy from Mursi, though Egypt's first freely elected leader, whose Muslim Brotherhood inspired Hamas's founders, has been careful to stick by the 1979 peace deal with Israel struck by Cairo's former military rulers.
Clinton, who broke off from an Asian tour with President Barack Obama and assured Netanyahu of "rock-solid" US support for Israel's security, spoke of seeking a "durable outcome" and of Egypt's "responsibility" for promoting peace.
She repeated international calls for the kind of lasting, negotiated, comprehensive Israeli-Palestinian settlement that has eluded the two peoples for decades - something neither of the two warring parties seems seriously to be anticipating.
"In the days ahead, the United States will work with our partners here in Israel and across the region toward an outcome that bolsters security for the people of Israel, improves conditions for the people of Gaza and moves toward a comprehensive peace for all people of the region," Clinton said.
"It is essential to de-escalate the situation in Gaza. The rocket attacks from terrorist organisations inside Gaza on Israeli cities and towns must end and a broader calm restored.
"The goal must be a durable outcome that promotes regional stability and advances the security and legitimate aspirations of Israelis and Palestinians alike."
Netanyahu, who has appeared in no immediate rush to repeat the invasion of winter 2008-09 in which over 1,400 Palestinians died, said: "If there is a possibility of achieving a long-term solution to this problem with diplomatic means, we prefer that.
"But if not, I'm sure you understand that Israel will have to take whatever action is necessary to defend its people."
As Israeli aircraft have carried out hundreds of strikes on rocket stores, launchpads and suspected Hamas command posts since assassinating the head of its military wing a week ago, tens of thousands of Israeli soldiers have been preparing tanks and infantry units for a possible invasion.
During the night, explosions again rocked the city of Gaza and other parts of the Strip, while rockets from the enclave, some essentially home-made, others Iranian-designed and smuggled through tunnels from Egypt, landed in southern Israel.
One reached as far as Rishon Lezion, near Tel Aviv, on Tuesday, the latest to jar Israel's metropolis, long untroubled by Palestinian attacks. Another rocket fell close to Jerusalem, the holy city claimed by both sides in the conflict.
Medical officials in Gaza said 31 Palestinians were killed on Tuesday. An Israeli soldier and a civilian died when rockets exploded near the Gaza frontier, police and the army said.
Gaza medical officials say 138 people have died in Israeli strikes, mostly civilians, including 34 children. In all, five Israelis have died, including three civilians killed last week.
Obama, whose relations with the hawkish Netanyahu have long been strained, has said he wants a diplomatic solution, rather than a possible Israeli ground operation in the densely populated territory, home to 1.7 million Palestinians.
Israel's military on Tuesday targeted more than 130 sites in Gaza, including ammunition stores and the Gaza headquarters of the National Islamic Bank. Israeli police said more than 150 rockets had been fired from Gaza by the evening.
"No country would tolerate rocket attacks against its cities and against its civilians," Netanyahu said with U.N. Secretary-General Ban Ki-moon, who arrived in Jerusalem from talks in Cairo, at his side. "Israel cannot tolerate such attacks."
Critics have accused Israel of using disproportionate force that has killed civilians. Israel accuses Hamas of putting Gaza's people in harm's way by siting rockets among them.
Media groups have criticised attacks on Gaza media facilities. On Tuesday, three local journalists died in air strikes on their vehicles.
A building housing AFP's bureau was bombed. The French news agency said its staff were unhurt. Israel's military said it had been targeting a Hamas intelligence centre in the tower.
Hamas executed six Palestinians accused of spying for Israel, who a security source quoted by Hamas Aqsa radio said had been "caught red-handed" with "filming equipment to take footage of positions". The radio said they had been shot.
Militants on a motorcycle dragged the body of one of the men through the streets.
A delegation of nine Arab ministers, led by the Egyptian foreign minister, visited Gaza in a further signal of heightened Arab solidarity with the Palestinians.