The Israeli military has carried out a new round of air strikes in Gaza, as Palestinian militants continue to fire rockets and mortars into Israel.
Two people were killed as two mosques and the Rafah border crossing were targeted, Palestinian officials said.
Meanwhile, efforts have been stepped up to end the six weeks of fighting that have left more than 2,100 people dead.
Egyptian mediators have reportedly proposed a new ceasefire deal that would see Gaza's crossings opened.
According to the initiative, aid and reconstruction materials would be allowed in while talks took place to resolve contentious issues that had prevented a long-term truce being agreed, a Palestinian official said.
'May take time'
The Israeli military said it had carried out 16 air strikes on "terror targets" in Gaza early on Monday, including rocket launchers and weapons manufacturing sites.
Two mosques in the north of the coastal territory were destroyed. Israeli officials said militants were using one of the mosques to store weapons and using the other as a meeting point.
Health ministry official Ashraf al-Qudra said a 42-year-old woman was killed by tankfire in the north of the territory, and that a 22-year-old man had died of his wounds on Monday after an air strike in Gaza City.
Another air strike damaged the departure lounge at the Rafah border crossing with Egypt. Separately, the Erez crossing with Israel was re-opened on Monday and then put "on freeze" after mortar fire in the area, Israeli officials said.
The Israeli military said militants in Gaza had fired at least 39 rockets and mortars at Israeli territory on Monday.
Israeli media reports said a woman was lightly injured when a rocket landed in Eshkol Regional Council.
Israel launched Operation Protective Edge on 8 July with the stated aim of ending rocket fire. It was later expanded to include the destruction of tunnels used by militants for cross-border attacks.
Prime Minister Benjamin Netanyahu said on Sunday the operation would "continue until its aims are achieved" and warned: "It may take time."
But on Monday morning, Egyptian officials were reported to have proposed a new ceasefire and the resumption of indirect talks in Cairo.
"There is an idea for a temporary ceasefire that opens the crossings, allows aid and reconstruction material, and the disputed points will be discussed in a month," a senior Palestinian official told the AFP news agency.
"We would be willing to accept this, but are waiting for the Israeli response to this proposal," he added.
A spokesman for Hamas, the Islamist movement that dominates Gaza, confirmed that efforts were continuing to reach an agreement, while an Islamic Jihad spokesman said the success of the talks depended on Palestinian demands being met.
Mr Netanyahu's spokesman said Israel's position that it would not negotiate while under fire had "not changed".
The Israeli government called home its representatives from Cairo six days ago after rocket fire from Gaza ended a nine-day ceasefire.
It has been seeking guarantees that Hamas and other factions in Gaza will be disarmed, while the Palestinians were demanding an end to the Israeli and Egyptian blockades, and the establishment of a seaport and airport.
At least 2,122 people, most of them civilians, have been killed in Gaza since Operation Protective Edge began, according to the Palestinian health ministry. The Israeli authorities say 64 Israeli soldiers have been killed, along with three Israeli civilians and a Thai national.