CAIRO — Emboldened by the rising power of Islamists around the region, the Palestinian militant group Hamas demanded new Israeli concessions to its security and autonomy before it halts its rocket attacks on Israel, even as the conflict took an increasing toll on Sunday.

After five days of punishing Israeli airstrikes on the Hamas-controlled Gaza Strip and no letup in the rocket fire in return, representatives of Israel and Hamas met separately with Egyptian officials in Cairo on Sunday for indirect talks about a truce.

The talks came as an Israeli bomb struck a house in Gaza on Sunday afternoon, killing 11 people, in the deadliest single strike since the conflict between Israel and Hamas escalated on Wednesday. The strike, along with several others that killed civilians across the Gaza Strip, signaled that Israel was broadening its range of targets on the fifth day of the campaign.

By the end of the day, Gaza health officials reported that 70 Palestinians had been killed in airstrikes since Wednesday, including 20 children, and that 600 had been wounded. Three Israelis have been killed and at least 79 wounded by unrelenting rocket fire out of Gaza into southern Israel and as far north as Tel Aviv.

Hamas, badly outgunned on the battlefield, appeared to be trying to exploit its increased political clout with its ideological allies in Egypt’s new Islamist-led government. The group’s leaders, rejecting Israel’s call for an immediate end to the rocket attacks, have instead laid down sweeping demands that would put Hamas in a stronger position than when the conflict began: an end to Israel’s five-year-old embargo of the Gaza Strip, a pledge by Israel not to attack again and multinational guarantees that Israel would abide by its commitments.

Prime Minister Benjamin Netanyahu of Israel stuck to his demand that all rocket fire cease before the air campaign lets up, and Israeli tanks and troops remained lined up outside Gaza on Sunday. Tens of thousands of reserve troops had been called up. “The army is prepared to significantly expand the operation,” Mr. Netanyahu said at the start of a cabinet meeting.

Reda Fahmy, a member of Egypt’s upper house of Parliament and of the nation’s dominant Islamist party, who is following the talks, said Hamas’s position was just as unequivocal. “Hamas has one clear and specific demand: for the siege to be completely lifted from Gaza,” he said. “It’s not reasonable that every now and then Israel decides to level Gaza to the ground, and then we decide to sit down and talk about it after it is done. On the Israeli part, they want to stop the missiles from one side. How is that?”

He added: “If they stop the aircraft from shooting, Hamas will then stop its missiles. But violence couldn’t be stopped from one side.”

Hamas’s aggressive stance in the cease-fire talks is the first test of the group’s belief that the Arab Spring and the rise in Islamist influence around the region have strengthened its political hand, both against Israel and against Hamas’s Palestinian rivals, who now control the West Bank with Western backing.

It also puts intense new pressure on President Mohamed Morsi of Egypt, a former leader of the Muslim Brotherhood who was known for his fiery speeches defending Hamas and denouncing Israel. Mr. Morsi must now balance the conflicting demands of an Egyptian public that is deeply sympathetic to Hamas and the Palestinian cause against Western pleadings to help broker a peace and Egypt’s need for regional stability to help revive its moribund economy.

 But Moussa Abu Marzouk, a senior Hamas official who was allowed to settle in Cairo after the ouster of President Hosni Mubarak, predicted a different outcome. In an interview at the same time, he said that if another conflict broke out with Israel, the moderate Islamist politicians around the region like the Egyptian Brotherhood would have to line up with the militants in Gaza.

“The position of all Islamists in the region will be that of Hamas,” Mr. Abu Marzouk said, “not the other way around.”

Israeli officials are conducting their side of the cease-fire talks through the contacts in Egyptian intelligence with whom they worked during Mr. Mubarak’s rule. Officials said their main focus was on ending the threat of rocket fire from Gaza, whether by diplomatic or military means.

Dan Meridor, the Israeli intelligence minister, said on Israeli television that the government would wait for Hamas “to stop firing” before it would negotiate a long-term cease-fire. In the meantime, he said, Israel would do “whatever it takes” to eliminate Hamas’s ability to fire rockets, potentially including an incursion into Gaza.

In his first public comments on Gaza since the latest violence broke out, President Obama said in Bangkok early Monday that he supported Israel’s right to take action in Gaza but that he was trying to defuse the conflict.

“We are actively working with all the parties in the region to see if we can end those missiles being fired without further escalation of violence in the region,” Mr. Obama said, noting that he had spoken with Mr. Netanyahu several times, as well as with Mr. Morsi and Prime Minister Recep Tayyip Erdogan of Turkey. “We’re going to have to see what kind of progress we can make in the next 24, 36, 48 hours,” Mr. Obama added.

As the conflict has intensified, so has diplomatic pressure on Israel to restrain its military campaign. William Hague, the British foreign minister, said in a television appearance on Sunday that he and Prime Minister David Cameron “stressed to our Israeli counterparts that a ground invasion of Gaza would lose Israel a lot of the international support and sympathy that they have in this situation,” The Associated Press reported.

While the Israelis talked to their longtime contacts in Egyptian intelligence, Mr. Morsi’s office worked through its own channels of communication with Hamas, and Mr. Morsi himself met on Sunday with Hamas’s top leader, Khaled Mashaal.

Mr. Fahmy, of Mr. Morsi’s Freedom and Justice Party, insisted Sunday that Israel was to blame for starting the current round of violence by killing Hamas’s top military leader, and that Israel would have to act to end it. “Now we’re exerting pressure to stop the fighting on both sides, but we can’t pressure the victim while the perpetrator isn’t even ready to settle,” he said.

Mr. Morsi, speaking Saturday night at a joint news conference with Mr. Erdogan, accused Israel of failing to abide by an earlier cease-fire with Hamas that Egypt had negotiated just a week earlier.

“There is a power imbalance,” Mr. Morsi said, noting the death tolls on each side: three Israelis killed by Hamas attacks during the five days of fighting, compared with more than 40 Palestinians killed by Israel, a figure that rose to 70 on Sunday.

“Israel is an occupying country, and international laws oblige occupiers with many things that Israel doesn’t abide by,” Mr. Morsi said. “If the situation was further escalated, or if a land invasion took place as Israelis have said, this would mean dire consequences in the region, and we could never accept that, and the free world could never accept that.”

Still, Mr. Morsi may not have a free hand. He is a new president of a country in a fragile political transition away from military-dominated rule. He must maintain good relations with Egypt’s still-powerful army and intelligence services, which are deeply wary of Hamas.

He has already shown a willingness to snub Hamas in the interest of Egyptian security, by leading a campaign to shut down the tunnels used to smuggle goods and occasionally weapons into Gaza under the Egyptian border. “We are closing them every day,” he said with evident passion in a recent interview.

Others in the Egyptian government argued that President Morsi was gaining a new perspective on Hamas, and on what officials of the Egyptian Foreign Ministry have long said was the group’s pattern of sacrificing the lives of Gazans to Israeli military campaigns for little reason other than to burnish its claim to be the champion of resistance to the Israeli occupation. That status is a key to its hold on power, and an asset in its rivalry with Fatah, the Western-backed faction that controls the West Bank.

Still, in his appearance on Saturday, Mr. Morsi publicly blamed only Israel for the violence, and warned its government that the Arab Spring had changed the Middle East. “Everyone should remember, the peoples of the region are different than before,” he said. “The leadership in the region is different.”