In neighboring Egypt, President Mohammed Morsi hosted leaders from Hamas and two key allies, Qatar and Turkey, to seek a way to end the fighting.
"There are discussions about the ways to bring a cease-fire soon, but there are no guarantees until now," Morsi said at a news conference. He said he was working with Turkey, Arab countries, the U.S., Russia and western European countries to halt the fighting.
Israel launched the operation on Wednesday in what it said was an effort to end months of rocket fire out of the Hamas-ruled territory. It began the offensive with an unexpected airstrike that killed Hamas' powerful military chief, and since then has relentlessly targeted suspected rocket launchers and storage sites.
In all, 46 Palestinians, including 15 civilians, have been killed and more than 400 civilians wounded, according to medical officials. Three Israeli civilians have been killed and more than 50 wounded.
Israeli military officials expressed satisfaction with their progress Saturday, claiming they have inflicted heavy damage to Hamas.
"Most of their capabilities have been destroyed," Maj. Gen. Tal Russo, Israel's southern commander, told reporters. Asked whether Israel is ready to send ground troops into Gaza, he said: "Absolutely."
The White House said President Barack Obama was also in touch with the Egyptian and Turkish leaders. The U.S. has solidly backed Israel so far.
Speaking on Air Force One, deputy national security adviser Ben Rhodes said that the White House believes Israel "has the right to defend itself" against attack and that the Israelis will make their own decisions about their "military tactics and operations."
The White House, which like Israel considers Hamas a terrorist organization, also continued to support Israel. "We believe Israel has a right to defend itself, and they'll make their own decisions about the tactics they use in that regard," deputy national security adviser Ben Rhodes told reporters on Air Force One.
Despite the bruising offensive, Israel has failed to slow the barrages of rockets from Gaza.
The Israeli military said 160 rockets were launched into Israel on Saturday, raising the total number to roughly 500 since this week's fighting began. Eight Israelis, including five civilians, were lightly wounded Saturday, the army said.
Israel carried out at least 300 airstrikes on Saturday, the military said, and it broadened its array of targets. One air raid flattened the three-story office building used by Hamas' prime minister, Ismail Haniyeh. He was not inside the building at the time.
In southern Gaza, aircraft went after the tunnels that militants use to smuggle in weapons and other contraband from neighboring Egypt. Tunnel operators said the intensity of the bombing was unprecedented, and that massive explosions could be heard kilometers (miles) away, both in Gaza and in Egypt.
The operators, speaking on condition of anonymity because of the illicit nature of their business, said they cannot approach the tunnel area to assess the damage, but the blasts appeared to be more powerful than in Israel's last major push to destroy the tunnels during a previous offensive four years ago. The tunnels are a key lifeline for Hamas, bringing in both weapons and supporting a lucrative trade that helps fund the group's activities.
Missiles also smashed into two small security facilities and the massive Hamas police headquarters in Gaza City, setting off a huge blaze that engulfed nearby houses and civilian cars parked outside, the Interior Ministry reported. No one was inside the buildings.
Air attacks knocked out five electricity transformers, cutting off power to more than 400,000 people in southern Gaza, according to the Gaza electricity distribution company. People switched on backup generators for limited electrical supplies.
Hamas has unveiled an arsenal of more powerful, longer-range rockets this week, and for the first time has struck at Israel's two largest cities, Jerusalem and Tel Aviv. Both cities, more than 70 kilometers (45 miles) from Gaza, had previously been beyond rocket range.
In a psychological boost for Israel, a new rocket-defense system known as "Iron Dome" knocked down a rocket headed toward Tel Aviv, eliciting cheers from relieved residents huddled in fear after air raid sirens sounded in the city.
Associated Press video showed a plume of smoke following an intercepting missile out of a rocket-defense battery deployed near the city, followed by a burst of light overhead as it struck its target.
Police said a second rocket also targeted Tel Aviv. It was not clear where it landed or whether it was shot down. No injuries were reported. It was the third straight day the city was targeted.
Israel says the Iron Dome system has shot down some 250 of 500 rockets fired toward the country this week, most in southern Israel near Gaza.
Saturday's interception was the first time Iron Dome has been deployed in Tel Aviv. The battery was a new upgraded version that was only activated on Saturday, two months ahead of schedule, the Defense Ministry said.
Israel has vowed to stage a ground invasion, a scenario that would bring the scale of fighting closer to that of a war four years ago. Hamas was badly bruised during that conflict but has since restocked its arsenal with more and better weapons. Five years after seizing control of Gaza, it has also come under pressure from smaller, more militant groups to prove its commitment to fighting Israel as it turns its focus to governing the seaside strip.
Israel's Defense Minister Ehud Barak has authorized the emergency call-up of up to 75,000 reserve troops ahead of a possible ground offensive. Israel has massed thousands of troops and dozens of tanks and armored vehicles along the border in recent days.
Egypt, which is led by Hamas' parent movement, the Muslim Brotherhood, has been spearheading efforts to forge a cease-fire. Morsi has vowed to stand strong with the people of Gaza and this week recalled Cairo's ambassador from Israel to protest the offensive.
Quietly, though, non-Muslim Brotherhood members in Morsi's government are said to be pushing Hamas to end its rocket fire on Israel. Morsi is under pressure not to go too far and risk straining ties with Israel's ally, the United States.
The Hamas website said Saturday that its leader, Khaled Meshaal, met with the head of Egyptian intelligence for two hours Saturday in Cairo, a day after the Egyptian official was in the Gaza Strip trying to work out an end to the escalation in violence.
Hamas has not immediately accepted Egypt's proposal for a cease-fire, but the group's website said it could end its rocket fire if Israel agrees to end "all acts of aggression and assassination" and lift its five-year Israeli blockade on Gaza. Egypt will present the Hamas position to Israeli officials.
Israeli officials say they are not interested in a "timeout," and want firm guarantees that the rocket fire, which has paralyzed life in an area home to 1 million Israelis, finally ends. Past cease-fires have been short lived.
Israeli Prime Minister Benjamin Netanyahu said he spoke with the leaders of Britain, Poland, Portugal, Bulgaria to press his case. "No government in the world would allow a situation where its population lives under the constant threat of rockets," Netanyahu told them, according to a statement from his office.
The diplomatic activity in Cairo illustrated Hamas' rising influence in a changing Middle East. The Arab Spring has brought Islamists to power and influence across the region, helping Hamas emerge from years of isolation.
Morsi warned that a ground operation by Irael will have "repercussions" across the region. "All must realize the situation is different than before, and the people of the region now are different than before and the leaders are different than before," he said at a joint press conference with Turkey's Islamist prime minister, Recep Tayyip Erdogan. Erdogan, like Morsi, leads an Islamist government that has chilly diplomatic ties with Israel.
On Friday, Morsi sent his prime minister to Gaza on a solidarity mission with Hamas. And on Saturday, Tunisia's Foreign Minister Rafik Abdessalem visited Gaza as well.