The new round of ceasefire talks is being held through Qatari and US mediators in Doha, and began on Saturday afternoon local time.
Israel is yet to respond publicly to the proposed deal, but said prior to the talks that it would not withdraw troops from Gaza or commit to an end to the war.
The proposal would not include these elements, the BBC understands.
Israel’s military announced the launch of a new offensive named “Operation Gideon’s Chariots” earlier on Saturday, amid the deadliest wave of strikes in Gaza in months.
At least 300 people have been killed since Thursday, rescuers say, including at hospitals and refugee camps in the north and south of the Strip.
Thousands have died since Israel resumed strikes on 18 March, following the collapse of a fragile ceasefire which lasted two months.
Aid agencies say Gaza’s grievous humanitarian situation has also worsened, as Israel has been blocking supplies of food and other aid from entering the territory for 10 weeks.
Israel’s Prime Minister Benjamin Netanyahu earlier this month promised a major military escalation in the war to occupy and control swathes of Gaza, force the Palestinian population to the south of the territory, and “destroy” Hamas.
Speaking from inside Gaza, journalist Ghada Al Qurd told the BBC’s Newshour programme there had been lots of “airstrikes, shellings, drones, shooting and even exploding, in the north and east.”
“It’s terrifying and horrible,” she said.
She said her family had only been having one meal a day, due to the scarcity and spiralling cost, and accused Israel of “using food as a weapon” – an allegation UN officials have also made in recent weeks.
