By Nidal al-Mughrabi and Alexander Cornwell

CAIRO/JERUSALEM (Reuters) -Hamas said on Wednesday it was studying what U.S. President Donald Trump called a "final" ceasefire proposal for Gaza but that Israel must pull out of the enclave, and Israeli leader Benjamin Netanyahu said 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.

Israeli Prime Minister Benjamin Netanyahu 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, U.S. 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 leaders believe that, with Iran weakened, other countries in the region have an opportunity to forge ties with Israel.

'SOME POSITIVE SIGNS'

Foreign Minister Gideon Saar said Israel 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.

Of 50 hostages held by Hamas, about 20 are believed to be still alive.

Israeli opposition leader Yair Lapid posted that his party could provide a safety net if any cabinet members opposed a deal, effectively pledging not to back a no-confidence motion in parliament that could topple the government.

At the end of May, Hamas had said it was seeking amendments to a U.S.-backed ceasefire proposal. Trump's envoy, Steve Witkoff, said this was "totally unacceptable."

That proposal involved a 60-day ceasefire and the release of half the hostages held by Hamas in exchange for Palestinian prisoners and the remains of other Palestinians; Hamas would release the remaining hostages as part of a deal that guarantees the end of the war.

"Israel has agreed to the necessary conditions to finalize the 60 Day CEASEFIRE, during which time we will work with all parties to end the War," Trump posted on Tuesday, without specifying the conditions.

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

Gaza health authorities said Israeli gunfire and military strikes had killed at least 139 Palestinians in northern and southern areas in the past 24 hours, and the Israeli military ordered more evacuations late on Tuesday.

Among those killed was Marwan Al-Sultan, director of the Indonesian Hospital in northern Gaza, in an airstrike that has also killed his wife and five children, medics said.

The Israeli military said it had targeted a "key terrorist" from Hamas in the Gaza City area. It said it was reviewing reports of civilian casualties and that the military regretted any harm to "uninvolved individuals" and takes steps to minimise such harm.

Hamas fighters stormed into Israel on October 7, 2023, and killed 1,200 people, most of them civilians, and took 251 hostages back to Gaza, according to Israeli tallies.

Israel's subsequent military assault has killed more than 57,000 Palestinians, most of them civilians, according to the Gaza health ministry, displaced almost all the 2.3 million population and caused a humanitarian crisis.

(Reporting by Nidal al-Mughrabi and Alexander Cornwell; Additional reporting by Hussam al-Masri, Charlotte Greenfield and Howard Goller; editing by Philippa Fletcher and Timothy Heritage)