Greek Prime Minister Antonis Samaras said: “It’s not only the future of our country, but the stability of the entire eurozone [that is at stake].”
France’s finance minister insisted they were a “whisker” away from a deal.
The chairman of the Eurogroup, Jean-Claude Juncker, said ministers would meet again next week.
“Greece did what it had to do, and what it had pledged to do… whatever technical difficulties in finding a technical solution do not justify any negligence or delay,” Mr Samaras said.
Greece needs the next tranche of its second bailout worth 130bn euros ($166bn; £104bn) to avoid insolvency.
The eurozone “would be threatened if we did not reach” a deal, French Finance Minister Pierre Moscovici said, before adding that “we are very close to a deal.”
“We have observed that Greece had made considerable efforts,” he told Europe 1 radio.
The eurozone finance ministers have been considering ways of reducing Greece’s public debt, which is projected to rise to 189% of gross domestic product (GDP) by next year.
“We believe that, eventually, eurozone leaders will agree on a deal to cut Greek debt substantially,” said Martin Koehring of the Economist Intelligence Unit.
“It is not in their interest to push Greece out of the eurozone over technical disagreements. A much more likely cause for a Greek disorderly default and euro exit would be domestic political developments in Greece, highlighted by rising political instability and social unrest.
“However, our assessment remains that there is a 40% probability that Greece leaves the eurozone within the next five years.”
The country’s bailout programme aims to get debt down to 120% of GDP by 2020.
There has been disagreement among the ministers and the International Monetary Fund, Greece’s other bailout creditor, on how to make the country’s debt manageable.
“The Eurogroup has had an extensive discussion and made progress in identifying a consistent package of credible initiatives aimed at making a further substantial contribution to the sustainability of Greek government debt,” Jean-Claude Juncker said in a statement.
Schroders economist Virginie Maisonneuve: “Clearly we’ll have more negotiation”
The eurozone ministers favour giving Greece an extra two years, to 2022, to bring its debt to 120% of GDP, but the IMF has resisted that extension.
Although the meeting wrapped up in the early hours of Wednesday morning in Brussels without a conclusion, earlier on Tuesday night there had been optimism a deal would be reached.
The French finance minister Pierre Moscovici said: “I have the impression that a political agreement is within reach.”
The managing director of the IMF, Christine Lagarde, insisted: “We’re going to work very constructively to see if we can find a solution for Greece. That’s what really is our goal, our purpose and our mission.”
So far, Greece has received nearly 149bn euros (£119bn; $191bn) from the eurozone and the International Monetary Fund, out of 240bn euros that has been approved in two bailout loans.