The Greek government has been told it will have to pass a series of sweeping reforms by Wednesday before talks with eurozone leaders on another bailout can begin.
Greece has requested a new loan of €53.5bn (£38.5bn) over three years to avoid financial collapse and a potential exit from the single currency.
Officials in Brussels say the figure may have to be much higher - up to €86bn (£61.6bn) - and they have insisted on tougher austerity measures at an emergency summit that has yet to produce an agreement after more than 12 hours of talks.
However, a "compromise" on the bailout deal has reportedly been proposed by the leaders of Greece, Germany, France and the European Union, a source told the AFP news agency.
This will now be put to the rest of the eurozone leaders, the source added.
The leaders earlier said Athens must push six measures, including tax and pension reforms, through parliament to restore trust before discussions start, a draft document showed.
But in a sign of division among the leaders, the document sent to them by finance ministers also included an option for Greece to temporarily leave the euro if no deal was agreed - a potential "time-out".
Not all ministers have endorsed this idea, which a senior EU source said was illegal and would not make it into the summit statement.
The Greek government has apparently agreed to a series of demands it previously criticised, including allowing creditors' officials to have a base in the capital Athens.
In the document, Greece agreed to carry out "ambitious" pension and tax reforms and "significantly" increase its privatisation programme.
And it has also accepted it will push reforms into law this week on a first set of measures as it tries to avoid a "Grexit".
But stumbling blocks remain.
The Greek government believes some of the measures demanded by creditors, such as a requirement to put €50bn (£35.8bn) worth of state-owned assets into a fund for subsequent sell-off, are intended to "humiliate" them.
One diplomat said it was tantamount to turning the country into a "German protectorate", stripping it of more sovereignty.
If Greece meets the conditions, the German parliament would meet on Thursday to allow Chancellor Angela Merkel and finance minister Wolfgang Schaeuble to open talks on a new loan.
Then Eurogroup finance ministers would meet on Friday or at the weekend to formally launch the negotiations.
The draft said Greece needs €7bn (£5bn) by 20 July, when it must make a payment to the European Central Bank, and a total of €12bn (£8.6bn) by mid-August when another ECB payment is due.
Greece's proposals for a new bailout, backed by parliament, are aimed at offering to bring in tax rises and spending cuts in return for another loan.
Greece has already had two bailouts worth €240bn (£172.2bn) from eurozone countries and the International Monetary Fund since 2010.
The eurozone talks were not about whether to agree a new bailout, but rather over whether to allow further negotiations over the Greek rescue package.
Speaking ahead of the talks, Greece's Prime Minister Alexis Tsipras said: "We can reach an agreement tonight if all parties want it."
But Mrs Merkel cautioned there would not be a deal later on Sunday "at any price", and she said trust in debt-laden Athens had been lost, warning of "tough" negotiations ahead.
Meanwhile, French President Francois Hollande said he was going to do everything he could to keep Greece in the euro.