Hamas said on Wednesday it was studying what US President Donald Trump called a "final" ceasefire proposal for Gaza but that Israel must pull out of the enclave, while Israeli leader Benjamin Netanyahu insisted that Hamas would be eliminated.

Trump said on Tuesday Israel had agreed to the conditions needed to finalise a 60-day ceasefire with Hamas after a meeting between his representatives and Israeli officials.

In a statement, the Palestinian militant group said it was studying new ceasefire offers received from mediators Egypt and Qatar but that it aimed to reach an agreement that would ensure an end to the war and an Israeli pullout from Gaza.

The Israeli prime minister called for the elimination of Hamas in his first public remarks since Trump's announcement.

"There will not be a Hamas. There will not be a 'Hamastan'. We're not going back to that. It's over," Netanyahu told a meeting hosted by the Trans-Israel pipeline.

The two sides' statements reiterated long-held positions, giving no clues as to whether or how a compromise agreement could be reached.

"I hope it would work this time, even if for two months, it would save thousands of innocent lives," Kamal, a resident of Gaza City, said by phone.

Others questioned whether Trump's statements would deliver long-term peace.

"We hope he is serious like he was serious during the Israeli-Iranian war when he said the war should stop, and it stopped," said Adnan Al-Assar, a resident of Khan Younis in Gaza's south.

There is growing public pressure on Netanyahu to reach a permanent ceasefire and end the nearly two-year-long war, a move opposed by hardline members of his right-wing ruling coalition.

At the same time, US and Israeli strikes on nuclear sites in Iran and ceasefire agreed on in last month's 12-day Israel-Iran air war have put pressure on Hamas, which is backed by Tehran.

Israeli Foreign Minister Gideon Saar said his country was "serious in our will" to reach a hostage deal and ceasefire.

"There are some positive signs. I don’t want to say more than that right now. But our goal is to begin proximity talks as soon as possible," he said while visiting Estonia.

A source close to Hamas said its leaders were expected to debate the proposal and seek clarifications from mediators before giving an official response. (Reuters)