— Private investors in Greek bonds would need to accept losses of at least 50 percent to stabilize Greece’s finances, according to a joint European and International Monetary Fund analysis that has put bankers and politicians at loggerheads over how to bail out the country once and for all.

The contribution is more than double what a banking industry group agreed to over the summer. But Greece’s cratering economy and public opposition to more taxpayer-supported loans is putting pressure on private investors to fund a growing gap in the country’s rescue program.

On Thursday, the Greek Parliament passed new austerity measures — as strikes and violent protests rocked Athens — in an effort to convince Europe that it is committed to getting its economy on track. The measures would cut 30,000 civil servants from government payrolls, raise taxes and whittle collective bargaining rights.

The debate over how to divide the costs of rescuing Greece is one of the central questions European officials hope to resolve at a weekend summit that comes almost two years to the day after a newly elected government in Athens admitted that the country’s finances were “off the rails.” The meeting will cover an array of complex issues that U.S. and other world officials believe threaten the global economy and are pressuring the Europeans to address.

No other issue rubs as raw a financial nerve as the divvying of responsibility for more than $300 billion in bonds, bonds that Greece — it is now almost universally acknowledged — cannot repay. The issue has touched off a fierce lobbying effort ahead of the summit, with European officials in continuous negotiations and a delegation from the Washington-based Institute of International Finance (IIF) dispatched to work the corridors of Europe’s byzantine bureaucracy.

On Wednesday evening, an unexpected gathering of top European officials and IMF Managing Director Christine Lagarde in Frankfurt failed to bridge a divide between Germany, which is footing much of Greek’s bill and supports larger private-sector losses, and France, which is more concerned about the impact of greater losses on its banks.

German Chancellor Angela Merkel and French President Nicolas Sarkozy had said they would introduce plans to stem Europe’s debt crisis and preserve the euro Sunday during the summit. But on Thursday, Merkel’s spokesman said that the presentation was canceled and that the two leaders would hold a second summit Wednesday.

Merkel also called off a speech to her nation’s Parliament in which she was to have outlined on Friday her proposals for a comprehensive restructuring of Greece’s debt, for bank recapitalization and for revising Europe’s economic governance.

In meetings at the massive Berlaymont complex in eastern Brussels, IIF Managing Director Charles Dallara has pressed the case that banks, having agreed to a 21 percent reduction in the worth of their Greek bonds in July, should not be asked to reopen the offer.

Individual banks have been floating proposals in e-mails and private meetings, according to European officials, and warning that major losses could cause a cutback in lending that would crimp the economy. The problems could grow larger, the officials say, if the bailout fans fears that other countries — such as Ireland, Portugal and especially Italy, with its major economy — might also need, or begin to demand, similar treatment to lighten their debts.

“European banks feel they are being held hostage by the sovereign debt crisis,” Guido Ravoet, chief executive of the European Banking Federation, said in a statement. He criticized European leaders’ efforts to pressure banks to take greater losses, saying that politicians should solve the crisis with “stronger governance and budgetary discipline.”

A chief reason that the issue has lingered for two years is the concern that a write-down of privately held Greek bonds could cause serious unintended consequences, such as prompting investors to run from Italian or Spanish bonds and causing a crisis in those countries.

At the outset, some analysts argued that Greece was insolvent, with debts that far outstripped its ability to pay, and needed a massive round of debt restructuring to sustain its economy.

That point of view was rejected. Key European officials such as the Central Bank president, Jean-Claude Trichet, said that allowing a default by any of the 17 euro zone countries would undermine the entire currency union. That strict “no default” approach carried great weight in the early months of the crisis. The IMF, while initially cautioning that Greece’s “debt dynamics” could spiral out of control, also signed off on efforts to keep the country afloat solely with public-sector loans and without debt restructuring.

But Greece’s economy continued to slide, and its officials fell behind on promises to cut spending and raise revenue through privatization and other efforts. Even a revised bailout program approved in July, which included the 21 percent reduction in the value of bonds, is now off track.

In an analysis prepared for the summit, IMF and European officials said that under the current program, Greece’s total outstanding debt will hit 180 percent of its annual economic output by next year. More disturbing, even with the government’s austerity measures and the agreed contribution by the private sector, the debt ratio would remain at that unsustainable level for years to come, said a European official who described the report. The document, which is central to this weekend’s discussion, was being distributed to finance ministers and other top officials Thursday.

To reverse the slide would require a deeper cut for private investors, stepped-up austerity in a nation already racked by strikes and deepening recession or more-generous loans from Greece’s European neighbors and the IMF.

“Everybody wants to push it on somebody else,” and each side has compelling arguments, said Andre Sapir, a senior fellow at the Brussels-based Bruegel think tank and former economic adviser to the European Commission. “There is no way to reduce the Greek debt without something unpleasant.”

The shocks could be social, financial and political. Demanding even greater austerity could intensify riots and protests in Greece and push the country into a full-blown depression. Private-sector losses could touch off a run against the debt of other European countries or spark a round of bank failures. Taxpayers in northern European countries are already vexed about the situation in Greece, and some analysts fear that any more bailouts could incite them to reject the gradual political integration that has been a hallmark of postwar Europe.

What has become clear, though, is that delay is expensive and has allowed the debt problems of a small nation to morph into a confidence-draining and systemic challenge to the euro region.

Some analysts are beginning to argue that Greece’s only choice is to renounce its debt altogether. The region’s problems have become so knotted and deep over the past year that even a 100 percent write-down would not free Athens of debt: Its budget is still in deficit, and the country would immediately need to borrow tens of billions of dollars to restart local banks that would be wiped out because they hold the bulk of the country’s loans.

Birnbaum reported from Berlin.