Israel launched air strikes across the Gaza Strip on Friday in response to Palestinian rockets fired after Egyptian-mediated talks failed to extend a 72-hour truce in the month-long war.
Egypt later called for a resumption of the ceasefire, saying only a few points remained to be agreed. Palestinian factions said they would meet Egyptian mediators later in the day, but there was no sign of any imminent deal.
There was also no immediate response from Israel. However, a government official said earlier that Israel would not negotiate with Palestinians while militants continued to unleash missiles.
As warning sirens sounded in southern Israel, the military said Hamas had fired at least 45 rockets on Friday morning and Israel's "Iron Dome" interceptor system had brought down two.
By resuming its attacks, Hamas appeared to be trying to put pressure on Israel, making clear it was ready to fight on to end a blockade of Gaza that both Israel and neighbouring Egypt have imposed on the impoverished enclave.
In the first casualties since hostilities resumed on Friday, Palestinian medical officials said a 10-year-old boy was killed in an Israeli strike near a mosque in Gaza City. An Islamic Jihad militant was killed in a later hit, local officials said.
In Israel, police said two people were injured by mortar fire from Gaza.
After a huge explosion in Gaza City, apparently from an air raid, a military spokesman said Israel had responded to Hamas rocket fire by launching air strikes at "terror sites" across the Gaza Strip.
"We will continue to strike Hamas, its infrastructure, its operatives, and restore security for the State of Israel," Lieutenant-Colonel Peter Lerner said in a statement.
Heavy civilian casualties and destruction during Israel's campaign against militants in packed residential areas of the Gaza Strip have raised international alarm over the past month, but efforts to prolong a ceasefire at talks in Cairo failed.
Israel had earlier said it was ready to agree to an extension as Egyptian go-betweens pursued negotiations with Israeli and Palestinian delegates.
Hamas spokesman Sami Abu Zuhri said Israel had rejected most Palestinian demands. "However, we did not close the door and will continue with the negotiations," he said.
The Palestinians had wanted Israel to agree in principle to demands which include lifting the blockade on the Gaza Strip, the release of prisoners and the opening of a sea port, but this had been rebuffed, Abu Zuhri said.
Israel has shown little interest in easing its naval blockade of Gaza and controls on overland traffic and airspace, suspecting Hamas could restock with weapons from abroad.
In Cairo, the foreign ministry called on both sides "to return immediately to the ceasefire commitment and exploit the opportunity available to resume negotiations on the very limited sticking points that remain in the fastest possible time".
In Gaza, some families who had returned to their homes in the northern town of Beit Hanoun during the ceasefire gathered their belongings and headed back to the United Nations shelters where they had sought refuge over the past few weeks.
Beit Hanoun resident Yamen Mahmoud, a 35-year-old father of four, said: "Today I am fleeing again. I am not against resistance but we need to know what to do. Is it war or peace?"
Gaza officials say the war has killed 1,877 Palestinians, most of them civilians. Hamas said on Thursday it had executed an unspecified number of Palestinians as Israeli spies.
Israel says 64 of its soldiers and three civilians have died in the fighting that began on July 8, after a surge in Palestinian rocket salvoes into Israel.
It expanded its air and naval bombardment of the Gaza Strip into a ground offensive on July 17, and pulled its infantry and armour out of the enclave on Tuesday after saying it had destroyed more than 30 infiltration tunnels dug by militants.
Hamas's refusal to extend the ceasefire could further alienate Egypt, whose government has been hostile to the group and which ultimately controls Gaza's main gateway to the world, the Rafah border crossing.
The announcement that the truce would not be extended came a few minutes after it expired at 0500 GMT following lengthy talks that continued in Cairo through the night.
Palestinian negotiators from factions including Hamas, Islamic Jihad and Fatah left their hotel for talks with Egyptian intelligence at 9.30 p.m. on Thursday, returning more than six and a half hours later looking subdued and saying no deal had yet been reached.
Palestinian officials paced the lobby, speaking on their phones and holding meetings, trying to reach a final decision as the deadline approached. A source at Cairo airport said the Israeli delegation left shortly before the truce expired.