Israeli missiles hit one of the tallest buildings in Gaza early Tuesday, wounding at least 20 Palestinians.
The missiles struck the 13-story office and apartment tower after dark following Israeli warnings that an attack was coming.
Israeli airstrikes killed at least eight people Monday while the army said 80 Hamas rockets fell on Israeli territory, causing no casualties.
Egypt, which led failed peace talks earlier this month, is again appealing to Israel and Hamas to call a truce and reopen negotiations.
Cairo is proposing opening key crossings into Gaza to let in humanitarian aid and material to repair damaged buildings.
On Monday, Palestinian officials appeared to accept the Egyptian proposal. There was no word from the Israelis.
Nearly two months of Israeli airstrikes in response to Hamas has killed more than 2,100 Palestinians, most of them civilians, including close to 500 children. Sixty-four Israeli soldiers and four civilians have been killed.
There have been several cease-fires, and peace talks in Cairo. But violence resumed quickly when contentious negotiations broke down over Israel's demand that Hamas be disarmed and the Palestinians' call for an end to the Israeli blockade of Gaza.
Prime Minister Benjamin Netanyahu is telling Israelis to be prepared for the conflict to stretch into September. He also warns Gaza Palestinians to leave sites where Hamas militants are operating. He said any site used by Hamas is a target for Israeli forces.
Thousands of homes in the Gaza Strip have been destroyed or damaged in the conflict. Nearly 500,000 people have been displaced in the territory where Palestinians, citing Israeli attacks that have hit schools and mosques, say no place is safe.
Meanwhile, Egypt pressed the warring sides to stop fighting and resume negotiations to forge an enduring truce in Gaza. Cairo proposed opening key crossings into the blockaded territory and allowing humanitarian aid and reconstruction material to be taken into Gaza, where there is widespread destruction from the Israeli attacks aimed at Hamas.
Palestinian officials seemed willing to accept terms of the Egyptian proposal, but there was no immediate response from Israel.
There have been several temporary truces that halted fighting, the latest last week. But violence resumed quickly when contentious negotiations broke down over Israel's demand that Hamas be disarmed and the Palestinians' call for an end to the blockade and the opening of a seaport.
Israelis moving from border areas
The Israeli Defense Ministry said it is helping Israelis leave homes close to high-risk areas along the Gaza border in what is effectively the government's first large-scale voluntary evacuation effort in the nearly eight weeks of fighting.
The government has come under criticism for its inability to stop mortar fire and anger has risen since the death last week of a 4-year-old Israeli boy who was killed when a Palestinian mortar landed in Nahal Oz, a kibbutz near the Gaza border.
Mortars are usually fired from a short distance away, meaning they are less likely to be detected by Israeli's air-raid or anti-missile defense systems. The short-range weapon also gives people little time to scramble for cover.
Livnat Ginzbourg, a spokeswoman for Israeli communities along the Gaza border, told The Associated Press about 100 families were leaving their homes on Monday, following a similar number the previous day.
Ginzbourg told the AP that Monday's evacuation was the first time the government was coordinating and financing temporary accommodation for all families wishing to flee.
Defense Ministry spokesman Jonathan Mosery stressed Israelis were not begin told to evacuate the border areas, but that the government was assisting Israelis who live up to five kilometers (3 miles) from the Gaza border by paying for accommodations in areas farther away.
The latest escalation in the conflict erupted last week after a six-day temporary truce collapsed. Hamas spokesman Sami Abu Zuhri said there were efforts underway Monday to reach an extended cease-fire agreement, but neither side seemed to be easing its attacks.
Hamas, the dominant movement in the Gaza Strip, has said it will not stop fighting until the eight-year Israeli-Egyptian blockade on the enclave of 1.8 million people is lifted.
"There are efforts and calls in order to reach a cease-fire agreement but until this moment there is nothing to that effect. And until the aggression is stopped and the Palestinian demands are met, the resistance will continue protecting the Palestinian people in confronting the ongoing Israeli aggression," Zuhri said.
Both Israel and Egypt view Hamas as a security threat and are demanding guarantees that weapons will not enter the economically crippled territory.
Israel recalled its negotiators from Cairo after a cease-fire collapsed.
In Tehran, a commander in Iran's Revolutionary Guard Corps, General Amir-Ali Hajizadeh, said Iran would "accelerate" the arming of the Palestinians in retaliation for what it claims was an Israeli spy drone deployed for surveillance of an Iranian nuclear enrichment site.
Iran says it shot down the drone Saturday. Iranian state television Monday showed pictures purported to be pieces of the destroyed aircraft, although no Israeli markings were evident. Hajizadeh said the drone was not flown from Israel, but "from a country in the region."
Some information for this report provided by Reuters and AP.