Agreement on a 130-billion-euro rescue package with strict conditions would draw a line under months of uncertainty that has shaken the currency bloc, and avert an imminent bankruptcy.
After seven hours of talks, senior officials said ministers had found ways to cut Greece's debt to between 123 and 124 percent of gross domestic product by 2020, but were pressing for more. Negotiators for private bondholders had offered to accept a bigger loss to help plug the funding gap.
A report prepared for ministers by EU, European Central Bank and IMF experts, obtained exclusively by Reuters, said Greece would need extra relief to cut its debts to the official target of 120 percent of GDP by 2020.
If Athens did not follow through on economic reforms and savings, its debt could hit 160 percent by that date.
"Given the risks, the Greek program may thus remain accident-prone, with questions about sustainability hanging over it," the 9-page confidential report said.
The euro zone sources said national central banks could restructure Greek bonds held in their investment portfolios in the same way as private investors, cutting Athens' debt by another 3.5 percentage points.
If the ECB were to forego profits on its Greek holdings, that would raise another 5.5 percentage points of GDP, the report showed. However, the sources said some euro zone countries were reluctant to pursue this option.
Diplomats and economists say a deal may only delay a deeper default by a few months. A turnaround could take as much as a decade, a bleak prospect that brought thousands of Greeks onto the streets to protest against austerity measures on Sunday.
EU sources said the minister needed to agree new measures to find some 6 billion euros to make the financing work, given the ever-worsening state of the Greek economy.
An accord will enable Greece to launch a bond swap with private investors to help reduce and restructure Athens' vast debts, put it on a more stable financial footing and keep it inside the 17-country euro zone.
A senior euro zone source said the finance ministers were negotiating for private sector creditors to take a loss of at least 53.5 percent on the nominal value of Greek bonds under the debt swap, up from a previously agreed 50 percent loss.
Earlier in the day, French Finance Minister Francois Baroin said all the elements were in place to reach an agreement and Greek Finance Minister Evangelos Venizelos said he expected a deal.
"We expect today the long period of uncertainty - which was in the interest of neither the Greek economy nor the euro zone as a whole - to end," Venizelos said in a statement.
Dutch Finance Minister Jan Kees de Jager, the most outspoken of Greece's creditors, said the Netherlands could not approve the rescue package until Greece had met all its obligations. But the chairman of the Eurogroup, Jean-Claude Juncker, said Athens had met all the prior conditions demanded of it.
Finland, another stern creditor, signed a side-deal with Greece for Greek banks to provide collateral in cash and highly rated assets in return for Finnish loan guarantees, removing one long-running obstacle.
