With the death toll from nearly nine months of war mounting and conditions worsening daily for Gazans, both sides are under increased international pressure to agree a ceasefire.
Qatar, working with the United States, has been leading the mediation.
"We exchanged some ideas with the mediator brothers with the aim of stopping the aggression against our Palestinian people," a Hamas statement said.
The office of Israeli Prime Minister Benjamin Netanyahu and the Mossad intelligence service confirmed the new approach.
"Israel is evaluating the remarks and will convey its reply to the mediators," said an Israeli statement.
Tens of thousands flee S Gaza
Tens of thousands of Palestinians have fled southern Gaza since Israel issued an evacuation order, amid fears of a new offensive in the area which came under Israeli bombardment yesterday.
On Israel's northern front, Lebanon's Hezbollah said it fired 100 rockets at Israeli targets in retaliation for a strike that killed a senior commander, heightening fears of full-scale war between the longtime foes.
The United Nations said war had "unleashed a maelstrom of human misery" and that the latest evacuation order had plunged yet more Palestinians into "an abyss of suffering".
Ten days after Netanyahu said the war's "intense phase" was winding down, the Israeli military again rained down air strikes and artillery fire on the Shujaiya district.
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