GAZA CITY, Gaza Strip — Israel and Gaza’s Hamas rulers traded fire and tough cease-fire proposals Monday, and threatened to escalate their border conflict if diplomacy fails. No deal appeared near.

An Israeli airstrike targeting a Gaza media center killed a senior militant and engulfed the building in flames, while Gaza fighters fired 95 rockets at Israel, nearly one-third of them intercepted by an Israeli missile shield.

A total of 38 Palestinians were killed Monday. Two more Palestinians were killed in airstrikes past midnight, bringing the death toll since the start of Israel’s offensive to 111, including 56 civilians. Some 840 people have been wounded, including 225 children, Gaza heath officials said. Three Israeli civilians have been killed and dozens have been wounded.

Over the weekend, civilian casualties in Gaza rose sharply after Israel began targeting the homes of what it said were suspected militants. Two such strikes late Monday killed five people — a father and his 4-year-old twin sons in northern Gaza and two people in the south, medics said.

Jamal Daloo, who lost his wife, a son, four grandchildren and five other members of his family in an attack Sunday, sat in quiet mourning Monday next to the ruins of his home, his face streaked with tears.

“The international public opinion witnessed the facts,” he said, speaking as his 16-year-old daughter, Yara, was still missing under the rubble being cleared away by bulldozers. “This does not require my words.”

Egypt, the traditional mediator between Israel and the Arab world, was at the center of a flurry of diplomatic activity Monday. Egyptian intelligence officials met separately in Cairo with an Israeli envoy and with Khaled Mashaal, the top Hamas leader in exile.

Hamas wants Israel to halt all attacks on Gaza and lift tight restrictions on trade and movement in and out of the territory that have been in place since Hamas seized Gaza by force in 2007. Israel demands an end to rocket fire from Gaza and a halt to weapons smuggling into Gaza through tunnels under the border with Egypt.

With positions far apart on a comprehensive deal, some close to the negotiations suggested Egypt is first seeking a halt to fighting before other conditions are discussed. They spoke on condition of anonymity because the talks are in a sensitive stage.

Mashaal told reporters that Hamas would only agree to a cease-fire if its demands are met. “We don’t accept Israeli conditions because it is the aggressor,” he said. “We want a cease-fire along with meeting our demands.”

Mashaal also suggested that Israel’s threat of invading Gaza was simply a ploy. He said Israeli Prime Minister Benjamin Netanyahu “is waving the threat of a ground offensive and asking the world to pressure Egypt, Turkey and Qatar, so they pressure Hamas.”

“He wants to negotiate with us under fire to impose his conditions, pretending he is acting from a position of strength,” Mashaal said.

Israeli leaders have repeatedly threatened to widen the offensive, saying an invasion is an option. Israel has amassed troops on the Gaza border and begun calling up thousands of reservists.

Still, an Israeli official emphasized that Israel hopes to find a diplomatic solution.

“We prefer the diplomatic solution if it’s possible. If we see it’s not going to bear fruit, we can escalate,” he said, speaking on condition of anonymity because of the diplomatic efforts under way. He added that Israel wants international guarantees that Hamas will not rearm or use Egypt’s Sinai region, which abuts Gaza, for militant activity.

As part of global efforts to end the Gaza fighting, U.N. chief Ban Ki-moon arrived in Cairo on Monday and was to meet with Israeli President Shimon Peres on Tuesday.

The U.N. Security Council held closed-door consultations at the request of Russia, and  Ambassador Vitaly Churkin later accused one country of foot-dragging, implying it was the U.S.

Germany’s foreign minister was also headed to the region for talks with Israeli and Palestinian leaders. On Tuesday, Turkey’s foreign minister and a delegation of Arab League foreign ministers were to visit Gaza.

Hamas, an offshoot of the region-wide Muslim Brotherhood, is negotiating from a stronger position than four years ago, when Israel launched a three-week war on the militants in Gaza. At that time, Hamas was internationally isolated; now, the Muslim Brotherhood is in power in Egypt and Tunisia, and Hamas is also getting political support from Qatar and Turkey.

President Barack Obama and other Western leaders have blamed Hamas for the latest outbreak of fighting, saying Israel has a right to defend itself against rocket attacks. However, they have also warned Israel against sending ground troops into Gaza, a move that would likely lead to a sharp increase in the Gaza death toll.

Over the years, Israeli governments have struggled to come up with an effective policy toward Hamas, which is deeply rooted in Gaza, a densely populated territory of 1.6 million.

Neither Israel’s economic blockade of the territory nor bruising military strikes have cowed the Islamists, weakened their grip on Gaza or their ability to fire rockets at the Jewish state.

Instead, the two sides have observed informal cease-fires over the years, interrupted by flare-ups of violence.

Hamas has fired more than 1,000 rockets at Israel since the start of the latest offensive on Wednesday, kicked off by Israel’s assassination of the Hamas military chief.

Of the 95 rockets fired Monday, 29 of them intercepted by Israel’s U.S.-financed Iron Dome anti-missile battery, police spokesman Micky Rosenfeld said. Rockets landed in open areas of the southern cities of Beersheba, Ashdod and Ashkelon, and caused damage in a number of areas, including an empty school building in Ashkelon.

In Gaza, an Israeli airstrike on a high-rise building in Gaza City killed Ramez Harb, a senior figure in Islamic Jihad’s military wing, the Al Quds Brigades, the group said. Israel said the target was a command center for the group. A number of foreign and local news organizations have offices in the building, which was also struck on Sunday. A passer-by, a carpenter from Gaza’s tiny Christian community, was also killed, medics said.

And in central Gaza, four militants were killed in two separate strikes. In the air raid past midnight, Israeli aircraft struck the Islamic National Bank used to pay Hamas employees.

In the West Bank, Palestinian stone throwers protesting against Israel’s Gaza campaign clashed with Israeli soldiers in several locations Monday. In the city of Hebron, a 22-year-old man was killed by army fire and three other protesters were injured, doctors said. The army said soldiers opened fire after a masked man approached them and failed to stop.