The German government's previous insistence on what the finance minister called “fair burden sharing” had renewed market jitters by threatening to derail negotiations on a second rescue of Greece, which will be needed to avert another financing crisis next year.

The German compromise was one of several fresh moves in the long-running saga over Greece's finances that could restore confidence among both taxpayers and investors. Prime Minister George Papandreou on Friday named a new finance minister and other cabinet members at the end of a week of political instability and angry street protests.

Chancellor Angela Merkel and the French president, Nicolas Sarkozy, announced their agreement after a two-hour meeting in Berlin.

“We would like to have a participation of private creditors on a voluntary basis,” Mrs. Merkel said at a joint news conference with Mr. Sarkozy.

“This should be worked out jointly with the ECB,” she added. “There shouldn't be any dispute with the ECB on this.”

“This is a breakthrough,” Mr. Sarkozy said, referring to the softening of the German position.

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Wall Street and European stock markets turned positive on the news and the euro strengthened against the dollar, reversing its earlier decline. Premiums on Greek and other bonds declined after a weeklong rout, according to Reuters.

To stave off an imminent default, Greece needs the next installment of the 110 billion euro, or $155 billion, loan package it received a year ago. That amounts to 12 billion euros. Further out, Greece is most likely to need another bailout — estimated at up to 60 billion euros — because it won't be able to return to markets next year as initially planned.

The European Central Bank — which itself holds billions of euros in shaky Greek debt — has firmly opposed anything that could trigger what rating agencies call a “credit event,” or default. Mario Draghi, who has been nominated to succeed Jean-Claude Trichet as bank president, testified on Tuesday that the bank could accept including bondholders only if it were “entirely voluntary.”

Mrs. Merkel, who has been weakened politically by a series of local election defeats, now faces the potential for a rebellion in her center-right coalition over the concession.

In addition, like-minded countries that have backed Mrs. Merkel and her finance minister, Wolfgang Schäuble, including the Netherlands, Austria and Finland, could also still protest.

On the other side, countries like France, whose banks are the most exposed to Greece, and the ECB, which has been a buyer of last resort for Greek sovereign debt, are afraid of anything that smacks of default. Such a “credit event” could lead to damaging losses for banks and a freezing up of the global credit markets, such as followed the Lehman Brothers bankruptcy in 2008.

Officials with the European Union and the International Monetary Fund officials have expressed confidence that an agreement to release the 12 billion euros from the next loan installment could be made at a meeting of euro group finance ministers on Sunday night in Luxembourg, while the question of a second rescue package could be put off until July. But politically, any new rescue package depends on Greece pushing through additional savings to close a widening budget gap — a demand that provoked a government crisis in the country this week.

Another issue that will need to be resolved Sunday is the source of the financing for a second Greek bailout.

Mr. Draghi has indicated that one acceptable option is the so-called Vienna Initiative, named after a 2009 agreement under which international lenders agreed to roll over credit lines and maintain their exposure to Central and East European countries to carry them through the global financial crisis.

On Friday, Mrs. Merkel said the Vienna Initiative was a “good basis” for a solution. Mr. Sarkozy agreed. But neither gave details about how the private investors would work with the International Monetary Fund and the European Central Bank. They said they were waiting for the troika — the IMF, the ECB and the European Commission — to present its latest report on Greece's situation.

Amid suggestions that Germany was pushing to delay a decision on the second rescue until September, Mrs. Merkel said she wanted “a solution as quickly as possible,” and hoped the new package would be decided by next month.

Britain, which does not use the euro, was involved to a limited extent in a European Union fund that financed Portugal's bailout earlier this year. It is insisting that it does not want to take part this time. That is a political problem for Germany because its Parliament has called for the same fund to be used again, in addition to a newer, euro-zone-only fund.

The timing on any final accord to be discussed Sunday remains a sensitive issue, with the European Commission, the bloc's executive arm, pressing for a deal by a scheduled July 11 gathering of finance ministers in Brussels.

By then, it hopes the Greek Parliament will have agreed to adopt the austerity package on which new aid will be conditional.

The International Monetary Fund has been reluctant to release its portion of the next tranche of aid without a clear picture of how Greece will finance itself over the next 12 months. If a final decision on the second Greek bailout is delayed until September, then there is a risk that the IMF may have the same worries about releasing its next package of loans in the fall, provoking another crisis.

However, some officials believe that the political situation in Greece is so volatile that the new government officials might need time to rally sufficient support for the package of new austerity conditions it will need to implement.

On Friday, the Dutch prime minister, Mark Rutte, emphasized the importance of getting a solid agreement rather than a rushed one.

“I am content that France — like Germany and the Netherlands — recognizes the need for voluntary private sector involvement and I'm confident that ultimately we will reach an agreement. However, this is going to be an important decision and we must get it just right,” he said in a statement.

Stephen Castle reported from Brussels.