Hamas delegates arrived in Egypt’s capital Cairo on Sunday for crunch talks on a ceasefire with Israel.
A US official has been quoted by the BBC as saying that terms of a six-week pause had already been “more or less accepted” by Israel diplomats.
The US says the pause would see the release of more Israeli hostages and Palestinian prisoners.
However, Israeli media said the country was demanding details of which hostages were still alive for progress to be made in the talks.
The negotiations came after bloody fighting in Gaza saw 115 Palestinians killed and more than 750 injured in an attack on Thursday.
Pressure for a ceasefire has grown after warnings from aid organisations that there is a risk of famine in northern Gaza.There is a desperate need for more aid to reach Gaza's 2.3 million Palestinians, a quarter of whom the United Nations says face starvation.
US president Joe Biden has committed to providing humanitarian aid by air.
The possibility of talks has been set up by Egyptian diplomats and also counterparts from Qatar.
According to reports, Hamas delegates are ready to agree to a truce and this could be made public within the next 48 hours.
Mr Biden said last week that a deal could be reached as soon as Monday, although Washington has since rowed back from such a firm timetable.
The aim is to have it in place in time to halt the fighting for the Ramadan Muslim fasting month, which begins in a week.
The war began in October after Hamas fighters stormed through Israeli towns killing 1,200 people and capturing 253 hostages, according to Israeli tallies.
Since then, Israeli forces have killed more than 30,000 Palestinians, according to Gaza health authorities, with thousands more dead feared unrecovered under rubble.
Swathes of the Gaza Strip have been laid to waste, with many of the population made homeless.
