But finance ministers meeting in Brussels still have a few last issues to wrangle over, such as tighter controls over Greece's spending and further cuts to the country's debt load.
Greece needs to secure the (euro) 130 billion ($170 billion) bailout quickly so it can move ahead with a related (euro) 100 billion ($130 billion) debt relief deal with private investors, which needs to be in place quickly if Athens is to avoid a disorderly default on a bond repayment on March 20.
"I am of the opinion that today we have to deliver, because we don't have any more time," Jean-Claude Juncker, the prime minister of Luxembourg who also chairs the meetings of eurozone finance ministers, said as he arrived in Brussels.
An uncontrolled bankruptcy would likely force Greece to leave the 17-country currency union and return to its old currency, the drachma, further shaking its already beaten economy and creating uncertainty across Europe.
French Finance Minister Francois Baroin told Europe-1 radio that while details will have to be worked out, "the political commitments have been made" for the bailout package, Greece's second in two years.
"We now have all the elements of a deal: elements of a participation that remains voluntary for banks and private creditors, and for public creditors -- states, central banks," Baroin said ahead of the meeting.
Apart from the finance ministers of the 17 euro countries, the get-together will also be attended by the heads of Greece's other creditors: the International Monetary Fund, the European Central Bank and representatives of private holders of Greek debt.
Greece's Finance Minister Evangelos Venizelos, who arrived in Brussels Sunday night, said he was also optimistic that the new aid program could be agreed.
"Greece comes into today's Eurogroup meeting having fulfilled all the requirements for the approval of the new program," he said. "For Greeks, this is a matter of national dignity and a national strategic choice and no other integrated and responsible choice can be opposed to it."
The Greek parliament has faced down violent protests in Athens and nationwide strikes to approve the austerity and reform measures demanded by the eurozone. Its main political leaders have committed in writing to uphold the bailout terms even after general elections in April.
Despite Athens' efforts, however, several important elements of the deal remained unsolved.
To convince the rest of the eurozone that the new aid money won't be squandered, Greece is expected to be forced to set up a separate account that would ensure that it services its debt. This escrow account would give legal priority to debt and interest payments over paying for government services.
The idea behind such an account is that it would maintain pressure on Greece to stick to promised austerity and reform measures, without the eurozone risking the destabilizing effects of a default.
"What happened in the past cannot happen again, that billions (of euros) flow into Greece but get put into consumption there," said Maria Fekter, Austria's finance minister, as she headed into Monday's meeting.
But she cautioned that current proposals for administering the account themselves are expensive and those costs need to be reduced to make the plan feasible.
Eurozone finance ministry officials Sunday night concluded that only funds from the bailout would be funneled through the account and that Greece won't be required to pay in taxpayer money, German Finance Ministry spokeswoman Marianne Kothe said in Berlin, citing the current state of the negotiations.
The escrow account nevertheless would be an unprecedented intrusion into a sovereign state's fiscal affairs and could ultimately see Greece force to pay interest on its debt rather than its teachers, doctors or other government employees.
The second big outstanding issue is how to make sure that the current efforts to save Greece can actually bring the country's debts down to a manageable level in the longer term. In October, eurozone leaders and the IMF said that Greece's debt should be reduced to around 120 percent of annual economic output by 2020, from above 160 percent currently.
But a new report prepared by the European Commission, the ECB and the IMF concluded that the new bailout, Athens' spending cuts and a planned (euro) 100 billion debt relief from private investors would still leave Greece's debt at almost 129 percent of economic output by the end of the decade.
The eurozone is still discussing several ways to close this financing gap.
The Greek official said there appeared to be agreement on further reducing the interest rate on Greece's first, (euro) 110 billion bailout as well as having national central banks in the eurozone, which also hold some Greek bonds, participate in the debt relief.
But the official said that there was still no final decision from the ECB on whether it would be willing to transfer profits from its Greek bond holdings back to Athens. In contrast to the national central banks, who purchased Greek bonds as part of their overall investment strategies, the ECB bought its Greek holdings in an effort to ease market pressure on Athens.
A fourth option for closing the gap would be to demand further losses from Greece's private bondholders like banks and other investment funds. A current plan foresees private creditors to swap their old Greek bonds for new ones with half the face value, lower interest rates and much longer repayment periods.
But now there is a push for bondholders to also give up on an accrued interest payment of around (euro) 5.5 billion on their old bonds.
Another issue due to be discussed Monday is how much the IMF will contribute to the new rescue. The Washington-based fund has provided one-third of the bailouts for Ireland and Portugal and chipped in (euro) 30 billion for Greece's first (euro) 110 billion ($145 billion) rescue. But this time around, it looks as if the IMF will put up much less than one-third.
"The indication is that the figure will be rather low," a European Union official said Sunday, adding however that a final decision from the fund's board is still outstanding. IMF Managing Director Christine Lagarde will also be at the Brussels meeting Monday. The official was speaking on condition of anonymity because talks about the IMF contribution were not yet concluded.
The Greek government is expected to introduce in Parliament on Monday another two pieces of emergency legislation, including wage and pension cuts. There were scattered protests over the cuts in Athens on Sunday.