It is understood Israel has set a Thursday deadline for the Egypt-brokered talks to succeed.
The current conflict began last Wednesday when Israel killed Hamas's military commander, saying it wanted to end rocket attacks on Israel.
More than 110 Palestinians and three Israelis have been killed so far.
Israeli ministers met late into the night to discuss Egyptian truce proposals, as officials are due to restart talks in Cairo.
Israeli troops are massed along the border, raising fears of a ground offensive similar to that of 2008-09.
However, Israeli officials say that any possible ground invasion of Gaza has been put on hold while the ceasefire talks continue.
A spokesman told the BBC that Prime Minister Benjamin Netanyahu had telephoned US President Barack Obama.
"Israel wants talks to succeed but we're prepared to go into Gaza," the spokesman said.
US Secretary of State Hillary Clinton is due in Israel for crisis talks.
Mrs Clinton was to underline that "the best way to solve this is through diplomacy, so that you have a peaceful settlement that ends that rocket fire and allows for a broader calm in the region", said Mr Obama's Deputy National Security Adviser Ben Rhodes.
Meanwhile, the UN Security Council is debating a statement calling on both parties to stop attacks and address humanitarian needs in Gaza.
BBC correspondents in Gaza reported that it had been a relatively quiet night, although the Israeli military said it had carried out about 100 strikes mainly on smuggling tunnels and underground rocket-launching facilities.
Hamas officials say 110 Palestinians have been killed by Israeli raids since Wednesday. Three Israelis have died as a result of rocket attacks.
A father and his two sons - thought to be two and four - were killed overnight, Hamas health ministry officials said.
On Monday, among those killed was a commander of the Islamic Jihad militant group who died in an air strike on a Gaza City tower block housing media organisations.
On Tuesday morning, 29 rockets were fired from Gaza toward Israel - three managed to get through Israel's Iron Dome interception system, Israeli officials said.
One landed on a parked bus in Beersheba and a house in Ofakim. There were no casualties.
Regional risk
UN Secretary General Ban Ki-moon met Arab League Chief Nabil al-Arabi in Cairo on Tuesday.
"I'm here to appeal personally for an end to the violence and to offer my ongoing efforts to achieve a ceasefire. I'm also here to offer my heartfelt condolences to many civilians, especially to families of victims who were killed," Mr Ban told a joint news conference.
He is due to travel to Israel later on Tuesday. Mr Ban warned against a ground operation in Gaza.
"Further escalating the situation will put the entire region at risk," he said.
Egypt has been trying to broker a ceasefire with the help of Qatar and Turkey.
Turkish Foreign Minister Ahmet Davutoglu and Arab foreign ministers are expected to visit Gaza on Tuesday.
The content of the Egyptian plan is not known, but both Israel and Hamas have presented conditions.
Israel's demands include no hostile fire of any kind from Gaza and international efforts to prevent Hamas from rearming, while Hamas is demanding an end to the blockade on Gaza and "Israel's assassinations".
The BBC's Kevin Connolly in Jerusalem says the Cairo talks - in which Egyptian intelligence officials are meeting separately delegations from Israel and Hamas - appear to be at a crucial stage.
It is the sequence of events involving the key measures of the plan, such as a cessation of attacks and easing the blockade on Gaza, that may prove a stumbling block, he adds.
Meanwhile, UN Security Council members have been debating a draft statement on Gaza and are awaiting a response from their governments.
On Monday, US President Obama spoke to his Egyptian counterpart Mohammed Mursi and Mr Netanyahu, discussing ways to de-escalate the situation, the White House said.
Khaled Meshaal, political leader of the Islamist movement Hamas which controls Gaza, said that a truce was possible in Gaza - as was further escalation of the conflict.
Hamas seized control of Gaza in 2007, a year after winning a decisive victory in general elections. Israel withdrew from the strip in 2005 but maintains a blockade around it.
Israel, as well as the United States and the European Union, regards Hamas as a terrorist organisation.
Source: BBC
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