After talks in Ramallah with Palestinian President Mahmoud Abbas and a possible second meeting with Israeli Prime Minister Benjamin Netanyahu, Clinton will fly later today to Egypt, the main broker in efforts to end eight days of fighting and avert a possible Israeli ground offensive.
In Tel Aviv, at least 10 people were wounded when a bus was blown up in what the government called a terrorist attack. The incident, which touched off celebratory gunfire from militants in Gaza, threatened to complicate efforts to reach a ceasefire.
Israel's best-selling Yedioth Ahronoth newspaper said an emerging outline of a ceasefire agreement called for Egypt to announce a 72-hour ceasefire followed by further talks on long-term understandings.
Under the proposed document, which the newspaper said neither party would be required to sign, Israel would hold its fire, end attacks against top militants and promise to examine ways to ease its blockade of the enclave.
An Israeli political source said differences holding up a deal centred on a Hamas demand to lift the Gaza blockade completely and the kind of activity that would be allowed along the frontier, where Israeli troops often fire into the enclave to keep Palestinians away from an area near a border fence.
Hamas official Ezzat al-Rishq said the main stumbling block was "the temporary timeframe for a ceasefire that the Israelis want us to agree to".
The London-based Al Hayat newspaper said Israel wanted a 90-day period to determine "good intentions" before discussing Palestinian demands, a position the report said the groups have rejected.
Rishq said a short-term truce, whose proposed duration he did not disclose, "would only buy (Israel) time" until a general election in January and "we would have accomplished nothing in the way of a long-term truce".
Hamas sources said the group was also demanding control over Gaza's Rafah borders with Egypt, so that Palestinians could cross easily and Israeli guarantees to stop assassinating Hamas leaders.
Israel, one of the Hamas sources said, wanted a commitment from the group to stop smuggling through tunnels that run into Gaza under the Egyptian border. The tunnel network is a conduit for weapons and commercial goods.
Clinton, who flew to the Middle East last night from an Asian summit, said in her public remarks with Netanyahu that it was "essential to de-escalate the situation".
"The rocket attacks from terrorist organisations inside Gaza on Israeli cities and towns must end and a broader calm restored," she said.
Netanyahu told Clinton he wanted a "long-term" solution. Failing that, Netanyahu made clear, that he stood ready to step up the military campaign to silence Hamas' rockets.
"A band-aid solution will only cause another round of violence," said Ofir Gendelman, a Netanyahu spokesman.
While diplomatic efforts continued, Israel struck more than 100 targets in Gaza overnight, killing a Hamas gunman and destroying a cluster of Hamas government buildings.
Palestinians militants fired 31 rockets at Israel, causing no casualties, and Israel's Iron Dome interceptor system shot down 14 of them, police said.
Israel has carried out more than 1,500 strikes since the offensive began. Medical officials in Gaza said 139 Palestinians, most of them civilians, including 34 children, have been killed. Nearly 1,400 rockets have been fired into Israel, killing four civilians and a soldier, the Israeli military said.
In the West Bank city of Ramallah, Clinton held talks with Palestinian President Abbas, whose bid to upgrade the Palestinians' status at the United Nations, in the absence of peace negotiations with Israel, is opposed by Washington.
"Secretary Clinton informed the president that the US administration is exerting every possible effort to reach an immediate ceasefire and the president expressed his full support for this endeavour," said Palestinian chief negotiator Saeb Erekat.
"Once the Israelis accept to stop their bombardments, their assassinations, there will be a comprehensive ceasefire sustained from all parties," Erekat said.
A Palestinian official said that Egyptian intelligence officials would hold further discussions today with leaders of Hamas and the Islamic Jihad group.
"There may be a response from Israel that Egyptian mediators want to present to Hamas and Islamic Jihad leaders," the official said. "Let's be hopeful it would be something Palestinian factions can accept."
Like most Western powers, Washington shuns Hamas as an obstacle to peace and has blamed it for the Gaza conflagration. A UN Security Council statement condemning the conflict was blocked yesterday by the United States, which complained that it "failed to address the root cause," the Palestinian rockets.
Hamas for its part is exploring the opportunities that last year's Arab Spring has given it to enjoy favour from new Islamist governments, and from Sunni Gulf powers keen to woo it away from Shi'ite Iran.
It may count on some sympathy from Egypt's president, Mohamed Mursi, though that country's first freely elected leader, whose Muslim Brotherhood inspired Hamas' founders, has been careful to stick by the 1979 peace deal with Israel struck by Cairo's former military rulers.
In Jerusalem, Clinton assured Netanyahu of "rock-solid" US support for Israel's security, and praised Mursi's "personal leadership and Egypt's efforts thus far" to end the Gaza conflict and promote regional stability.
Along the Gaza border, Israeli tanks, artillery and infantry remained poised for a possible ground offensive in the densely populated enclave of 1.7 million Palestinians.
But an invasion, likely to entail heavy casualties, would be a major political risk for Netanyahu, who is currently favoured to win the upcoming Israeli election. More than 1,400 Palestinians were killed in Israel's three-week war in the Gaza Strip in the winter of 2008-9, prompting international criticism of Israel.