Jean-Claude Juncker, chairman of the meeting of finance ministers from the 17 countries that use the euro, said the talks, which lasted nearly 12 hours, will reconvene on Monday. It was the second consecutive meeting at which the ministers failed to agree on a deal, highlighting the depth of their divisions over how to handle Greece's huge debt problem without reaching more deeply into the pockets of their own taxpayers.
Juncker, however, said he was optimistic that a deal could be reached.
"We are very close to a result. We see no major stumbling block," he said. There are technical issues and calculations to be made in coming days, he said.
But Christine Lagarde, the managing director of the International Monetary Fund, which gives Greece bailout loans alongside the eurozone, sounded a more cautious note, saying only "we have narrowed the positions."
Investors reacted by selling the euro, which dropped from above $1.2810 to $1.2755 within a half hour. Stock markets in Asia surrendered early gains.
There has been disagreement among the ministers and the IMF on how to make Athens' debt manageable. The eurozone ministers are in favour of giving Greece an extra two years, to 2022, to bring its debt down to 120 per cent of gross domestic product from the 176 per cent forecast for this year. The IMF has resisted such an extension.
Agreement on this issue is needed for the group of creditors to pay Greece the next batch of its rescue loans, expected to amount to €44.6 billion ($57 billion). Greece the needs the money to avoid bankruptcy.
Greece has been relying since 2010 on international bailout loans, under terms supervised by the so-called troika — the IMF, the European Central Bank and the European Commission, which is the 27-country European Union's executive branch. Two weeks ago, Greece's coalition government narrowly succeeded in passing a €13.5 billion package of budget cuts, tax increases and reforms in order to secure the latest loan payment.
The main aim of the bailout program is to right the country's economy and get it to a point where it no longer relies on international aid and can independently raise money on the debt markets.
The question of debt sustainability is as important as it is divisive: If Greece's debts can't be reduced to a level where the country can afford to pay them, the billions of euros in bailout loans given to Greece will have been wasted.
One of the reasons Greece is not expected to reduce its debt to 120 per cent of GDP by 2020 is that its austerity program of spending cuts and tax hikes is hurting the economy, which faces a sixth year of recession in 2013.
To ease the impact on the economy, Greece's fellow eurozone countries agreed last week to grant Athens an extra two years until 2016 to implement its program of austerity reforms and budget deficit cuts. But this extension will also cost more money — another €32 billion through 2016, according to the troika — since the debt will have to be financed over a longer period.
Many eurozone countries are opposed to putting more money into Greece's aid program.
The eurozone countries could help pay for the extension by lowering the interest rates Greece has to pay for its rescue loans. They have also indicated that they would be willing to hand back to Greece profits they receive on Greek bonds held by the European Central Bank.
That may not be enough, though. Dutch finance minister Jeroen Dijsselbloem warned the extension could, in the end, cost new taxpayer money. "The chance is very large. I see it as my duty to make sure the damage is as limited as possible," he said.
The eurozone finance ministers will try on Monday to finally clinch an agreement on how to help Greece, but they will not be able to give final approval. Parliaments in some of the countries must first vote, after which the ministers will meet again on the issue.