Israel has stepped up its attacks on southern Gaza, targeting tunnels near the city of Rafah, and killing at least six Palestinians and injuring eight Israelis.
The Israeli military carried out a series of air strikes, targeting "a terror tunnel and a weapons facility" in the north and a rocket-launching site in the south, the army said on Monday.
Palestinian medics said the death toll stood at six, after an Islamic Jihad member named as Mohammed Shwikani, 20, was killed in an air strike near the northern Gaza town of Jabaliya.
Medics also reported finding the body of a second Jihad member, Mohammed Abed, 20, who was killed in another air raid east of Jabaliya.
Medics said 35 people were wounded, with Ashraf al-Qudra, spokesman for the health ministry, adding that 10 had undergone amputations.
He named the four civilians killed as 20-year-old Matar Abul Ata, 17-year-old Mohammed Harara, 15-year-old Ahmed Harara and 18-year-old Ahmed Dardasawy.
Gaza fighter retaliated by firing rockets and mortars into soutern Israel, causing no injuries or deaths.
Micky Rosenfeld, an Israeli police spokesman, said nine rockets had hit Israel, all of them fired after 0530 GMT.
"Seven were fired at the Negev (desert) region and two towards the Ashkelon (coastal) area," he said, adding that one had landed in the yard of a house in Netivot.
Israeli medical sources said 26 people had been treated for shock in the town.
Egyptian security sources said that Cairo stepped into its usual role as mediator on Sunday night to try to broker a truce.
Palestinian officials confirmed the truce initiative and said the two main fighter groups in the Gaza Strip, Hamas and the Islamic Jihad, had affirmed their readiness to observe it "provided Israel commits to doing the same."
Talk of a truce surfaced as Israel warned Gaza's Hamas rulers that it would strike back with "ever-growing intensity" for rocket attacks and that the conflict could escalate.
The threats from Israeli officials have raised the spectre of a broader Israeli operation against Gaza, along the lines of the devastating war on Gaza over the New Year of 2009.
"Hamas is responsible for the rocket fire and all other attempts to harm our soldiers and civilians, even when other groups participate," Defence Minister Ehud Barak said on Sunday. "And it is Hamas that will pay the heavy price."
In December 2008, just six weeks before general elections, Israel launched a 22-day war on Gaza that killed 1,400 Palestinians - half of them civilians - and 13 Israelis, 10 of them soldiers.
The latest violence comes ahead of Israeli elections in January. PM Binyamin Netanyahu warned earlier that Israel was prepared to escalate" its response to Palestinian rocket fire.
"The army is acting and will act forcefully against the terror organisations in the Gaza Strip. They are receiving strong blows from the army," Netanyahu told a meeting of his cabinet.
Tensions have been rising on the border since Thursday, when a 13-year-old Palestinian was shot dead by gunfire from an Israeli helicopter.
In a retaliation attack claimed by Hamas's armed wing, an Israeli soldier was wounded after explosives packed into a border tunnel were detonated.
On Saturday evening, armed groups fired an anti-tank missile at an Israeli jeep along the Gaza border, injuring four soldiers.
Israel attacked the coastal enclave with air strikes, shelling and artillery fire that killed six Palestinians and wounded 35 by Sunday night.