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Israel says it will push forward with the offensive until Hamas ends all rocket fire on southern Israel, and there are guarantees the militant group will stop smuggling weapons into Gaza through the porous Egyptian border.
Hamas has said it will only observe a cease-fire if Israel withdraws from Gaza.
"We will not allow our enemy to gain any political achievement from this war on Gaza," said Salah Bardawil, a Hamas envoy in Egypt.
Much of the ongoing diplomacy focuses on an area of southern Gaza just across the Egyptian border that serves as a weapons smuggling route, making Egypt critical to both sides in any deal.
Israel wants smuggling tunnels along the border sealed and monitored as part of any deal, and has bombed suspected tunnel sites throughout its campaign.
One resident, Khader Mussa, said he fled his house while waving a white flag as Israeli forces advanced. He spent the night huddling in the basement of a relative with 25 other people, including his pregnant wife and his parents.
"Thank God we survived this time and got out alive from here. But we don't know how long we'll be safe in my brother's home," Mussa, 35, said by telephone.
The Israeli military said it carried out dozens of airstrikes on squads of gunmen, rocket launching sites and smuggling tunnels along the Egyptian border.
The Gaza fighting has raised tensions around the region and galvanized anger toward Israel throughout the Arab world. On Tuesday, at least one gunman opened fire at an Israeli army patrol along the desert border between Israel and Jordan, the military said. There were no casualties, and Jordan said the claim was "baseless."
In the southern West Bank city of Hebron, the Israeli military said, a Palestinian was shot and injured after he tried to grab a gun from an Israeli soldier whose patrol stopped him for questioning. The man later died, according to an Associated Press reporter who saw his body.
Humanitarian concerns have increased amid the onslaught although some aid is getting through to Gaza during daily three-hour lulls declared by Israel to allow delivery of supplies.