After talks in Ramallah with Palestinian President Mahmoud Abbas, Clinton held a second meeting with Israeli Prime Minister Benjamin Netanyahu before travelling to Egypt for discussions with President Mohamed Mursi, whose country is the main broker in efforts to end eight days of fighting.
In Tel Aviv, targeted by rockets from Gaza that either did not hit the city or were shot down by Israel's Iron Dome interceptor system, 15 people were wounded when a commuter bus was blown up near the Defence Ministry and military headquarters.
Israel and the United States branded it a terrorist attack, and a White House statement reaffirmed Washington's "unshakeable commitment to Israel's security".
The explosion, which police said was caused by a bomb placed on the vehicle, touched off celebratory gunfire from militants in Gaza and threatened to complicate truce efforts.
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 Gaza, controlled by Hamas Islamists who do not recognise the Jewish state's right to exist.
Hamas, the report said, would pledge not to strike any Israeli target and ensure other Palestinian factions in the Gaza Strip also stop their attacks.
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".
While diplomatic efforts continued, Israel struck more than 100 targets in Gaza, including a cluster of Hamas government buildings, in attacks that medical officials said killed 10 people, among them a 2-year-old boy.
Palestinians militants fired more than 30 rockets at Israel, causing no casualties, and the Iron Dome interceptor system shot down 14 of them, police said.
Israel has carried out more than 1,500 strikes since the offensive began with the killing of a top Hamas commander and with declared aim of deterring Hamas from launching rocket attacks that have long disrupted life in its southern towns.
Medical officials in Gaza said 144 Palestinians, more than half of them civilians, including 36 children, have been killed in Israel's offensive. Nearly 1,400 rockets have been fired into Israel, killing four civilians and a soldier, the military said.
The London-based Al Hayat newspaper, citing sources in Hamas and Islamic Jihad, 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.
News of the Tel Aviv bus bombing, the first serious blast in Israel's commercial capital since 2006, caused oil prices to rise by more than $1 per barrel on concerns the Gaza crisis could lead to wider regional conflict that would disrupt oil flows.
Clinton, who flew to the region from an Asian summit, said after Tuesday's meeting 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.
Clinton earlier assured Netanyahu of "rock-solid" U.S. 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.
"As a regional leader and neighbour, Egypt has the opportunity and responsibility to continue playing a crucial and constructive role in this process. I will carry this message to Cairo tomorrow (Wednesday)," she said, pledging to work for a truce "in the days ahead".
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.
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 2008-9, prompting international criticism of Israel.
In the West Bank city of Ramallah, Clinton held talks with Palestinian President Abbas, reiterating U.S. opposition to his bid to upgrade the Palestinians' status at the United Nations. State Department spokeswoman Victoria Nuland said Washington believed "the best way to achieve statehood is through direct bilateral negotiations". Those talks collapsed in 2010 in a dispute over Israeli settlement building in the West Bank.
"Secretary Clinton informed the president that the U.S 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 with knowledge of Cairo's mediation told Reuters that Egyptian intelligence officials would hold further discussions on Wednesday with leaders of Hamas and the Islamic Jihad group.