Opposing Berlin is the International Monetary Fund, which has made more debt relief a condition of taking part in Greece's latest 86-billion-euro ($94-billion) bailout, its third since 2010.
Prime minister Alexis Tsipras is redoubling his push to reduce his country's debt burden after the economy fell back into recession unexpectedly at the start of the year.
"The Greeks have always been told, 'If you do this, then if necessary there will be debt relief,"' Gabriel told reporters in Berlin at a news conference with French Foreign Minister Jean-Yves Le Drian.
Pierre Moscovici said before the eurogroup meeting that a deal is "doable" and that it's time to "open a new phase, a new page" with regard to Greece.
German Chancellor Angela Merkel stands in the way of the debt relief, as she refuses to unlock more loans to Greece.
The eurozone's top official says a decision on whether Greece has done enough to get its hands on the next batch of rescue money that it's due from its bailout program could emerge later Monday.
Under the terms of the current Greek bailout program, the country has to enact a series of economic reforms in order to get the cash.
Without the deal no new loans can be granted to Athens, even though the bailout is now handled only by euro zone governments and Greece needs new credit to repay some €7.3 billion worth of maturing loans in July.
"I expect and I'm working on a deal today, but it will not be the end deal". That contrasts with guarded comments by Finance Minister Wolfgang Schaeuble, a Christian Democrat, who said euro-area finance ministers were discussing the matter on Monday in Brussels.
A group of north European countries led by Germany wants the International Monetary Fund to join for credibility reasons, believing the European Commission's approach towards Athens can be too lenient.
The country's GDP fell by 0.5 per cent year-on-year in the first three months of 2017, according to early estimates of growth by government statisticians.
French Finance Minister Bruno Le Maire, named last week, is joining European Union finance ministers for talks Monday and Tuesday expected to focus on Greece's debt problems. He said that "extra measures if required" would come after the bailout program expires next year.
Mr. Schaeuble later described reforms agreed by Greece as "remarkable" but said the Greek economy was not yet competitive and that Athens must press ahead with implementing its existing reforms-for-aid programme.
Successive Greek governments have slashed spending in return for bailout money to avoid bankruptcy.