Greece's prime minister has entered crunch bailout talks with creditors in Brussels.
Some officials dampened expectations of a breakthrough even though Greece is running out of cash and faces debt repayments as soon as Friday.
But French President Francois Hollande said the talks were at least heading in the right direction: "We are some days, not to say some hours away from a possible agreement."
Greece has been negotiating for four months with its creditors over what budget reforms it should make to get the €7.2 billion in loans that are left over in its bailout fund.
Today's meetings are part of a string of high-level diplomatic efforts to bring the negotiations to a successful end.
Ahead of his meeting with Commission head Jean-Claude Juncker, Greek leader Alexis Tsipras stressed the need for compromise.
"Today more than ever it is necessary for the (creditor) institutions and mainly for the political leadership of Europe to sign up to realism," Mr Tsipras said.
He spoke by teleconference with Mr Hollande and German Chancellor Angela Merkel ahead of his meeting with Mr Juncker, a Greek government official said.
The three agreed on the need for Greece to have lower primary surpluses - the budget balance without taking into account debt servicing - "and for a solution to be found quickly," the official said.
Lower primary surpluses than those initially demanded under Greece's five-year bailout have been one of Athens' main requests, although it appears to have been the easiest to overcome in the negotiations.
Sticking points appear to have been labour and pension reforms, as well as some changes to consumer tax.
The eurozone's top financial official, Jeroen Dijsselbloem, was also in Brussels and met with Mr Juncker before Mr Tsipras's arrival. He was less upbeat than others, saying: "We still have a lot of work to do."
That did not prevent Greek markets from rising on hopes of a deal, with the main stock index closing up 4.1%.
Time is pressing. Greece must repay €1.6 billion to the IMF this month alone. The first installment of just over €300 million is due on Friday, with other instalments on June 12, 16 and 19. Although Athens insists it intends to repay its debts, it is unclear how much longer it will be able to do so without outside help.
One option Greece could choose is to bundle this month's IMF payments into one on June 30, giving more time for negotiations. While allowed under IMF regulations, the option is rarely used.
Without bailout funds, Greece could eventually default on its debts and crash out of the euro, possibly pushing Europe and potentially the global economy into turmoil.
"We are working with high pressure" to find a solution, Mrs Merkel said in Berlin.
Mr Tsipras said he submitted his proposal to creditors on Monday night but gave no details of what fiscal reforms Greece was suggesting in return for its final bailout funds.
Greece needs the loans because it is locked out of the international bond markets as wary investors demand prohibitively high interest rates to lend it money. The country has not received bailout funds since August.
So far it has managed to scrape together enough to repay its debts by seizing the reserves of state enterprises, including municipalities, schools, embassies and hospitals. But those funds will not be enough to sustain the country through the summer.
Elected in January on promises to repeal the deeply resented austerity measures Greece had to impose in return for its five-year bailout, Mr Tsipras said his government had made concessions during the negotiations.
The 40-year-old premier faces dissent from hardliners from within his own radical left Syriza party, some of whom openly say they prefer a rupture in negotiations and even a euro exit, to capitulation on pre-election promises.
One of the most vocal critics of compromise has been Energy Minister Panagiotis Lafazanis.
"The logic of a deal at any cost ... is a logic of the country's total surrender to the worst neocolonial regime," Mr Lafazanis was quoted as telling the news website Real.gr.
"A deal will either be compatible with Syriza's progressive program, or it will not exist."