(Adds Haim Ramon comments, paragraph 13-14)

By Alastair Macdonald and Adam Entous

JERUSALEM, May 9 (Reuters) - Israeli Prime Minister Ehud Olmert defied a barrage of calls to resign on Friday after he admitted taking cash from an American businessman at the centre of a police inquiry into allegations of bribery.

Olmert, whose departure could disrupt U.S.-sponsored peace negotiations with the Palestinians, pressed on with his duties after telling the nation in a late-night address on Israel's 60th Independence Day that he would resign only if the attorney general could produce sufficient evidence to indict him.

Coalition allies have been conspicuously mute rather than offering public support. But the prime minister, who will host U.S. President George W. Bush next week, looked relaxed when speaking in Jerusalem to Canadians who raise funds for Israel.

He made only an oblique, passing reference to his troubles, saying: "I have enough political issues to deal with here."

"Millions of shekels -- cash in hand," screamed top-selling tabloid Maariv. All media splashed the story after police lifted a gag order imposed when Olmert was quizzed a week ago.

Yet Israelis have grown used to tales of corruption at the top and many noted that Olmert, who last year called himself "indestructible", has ridden out a series of other scandals.

Legal sources say police suspect that Olmert took hundreds of thousands of dollars from a New York Jewish financier over a decade. Morris Talansky was tagged "The Laundry Man" in coded records that investigators say were kept by Olmert's secretary.

The prime minister said Talansky funded two campaigns for Jerusalem mayor in the 1990s and for posts in the Likud party in 1999 and 2002. Olmert said his law partner Uri Messer handled the details; many see that as an attempt to divert blame if prosecutors pursue Olmert for failing to declare the money.

Israeli law broadly prohibits political donations of more than a few hundred dollars and senior figures, notably the son of Olmert's ally and predecessor Ariel Sharon, have been jailed for accepting larger or undeclared sums, even where prosecutors have failed to prove favours were done in return for bribes.

OLMERT'S SURVIVAL

"It is doubtful Olmert can survive," wrote Nahum Barnea, a senior columnist for Yedioth Ahronoth. "If not because of the Talansky affair ... then because of the cumulative effect of all the ongoing investigations against him."

Inside his Kadima party, where Foreign Minister Tzipi Livni is widely seen as Olmert's likely successor, one political source told Reuters: "He cannot survive. This is an investigation too far."

Others felt Olmert could survive for now. But some party officials fear that Kadima, founded by Sharon and Olmert in 2005, could break apart altogether under the strain.

Vice Premier Haim Ramon, a close Olmert ally, said the prime minister could continue to lead the country, as some of his predecessors did, while being investigated.

"Do we determine that whenever there is an investigation he needs to resign? Such a resignation would hinder the will of the people," Ramon said on Channel 2 television.

Many Israelis do not see an obviously popular successor.

"Olmert is a slick lawyer and he will get out of this affair as he did in the others," said social worker Adam Haisrael, 31. "He isn't fit to be prime minister but there's no one worthy to replace him. They're all seen as corrupt opportunists."

Analysts said Olmert may emphasise his peace-making and the security threat to Israel as he seeks to stay in office. On Friday, a mortar fired by Hamas militants from the Gaza Strip killed an Israeli -- the first such death in over two months.

The right-wing opposition Likud, led by Benjamin Netanyahu, is keen for an election that opinion polls suggest it could win: "Olmert and the Kadima government have no public legitimacy, no moral legitimacy," said Likud lawmaker Yuval Steinitz.

A key figure will be Defence Minister Ehud Barak, leader of Olmert's main coalition ally the Labour party. Barak has so far said little and is believed to be wary of bolting the alliance if that led to an election that would favour Netanyahu.

"I think for now Barak will bide his time," a Labour source said. "We could probably live with a Livni-led coalition.

"But whatever happens on this case, there is a sense of an accruing moral attrition that, I anticipate, will force Labour to reshuffle the deck at some point soon."

Palestinian peace negotiator Ahmed Qurie told Reuters he hoped any change in the Israeli leadership would not disrupt a peace process that Bush hopes can produce a deal on founding a Palestinian state before he leaves the White House in January.

But privately Palestinian officials acknowledged Olmert's woes could derail negotiations, especially if a snap election were called and the hawkish Netanyahu were returned to power. (Additional reporting by Dan Williams, Avida Landau, Rebecca Harrison and Ari Rabinovitch; editing by Andrew Roche)

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