By Jan Strupczewski
But no decisions are expected on Tuesday, euro zone sources close to the meeting said.
"There are no decisions expected today, it is just about looking for common ground in view of the meeting on June 20," said one euro zone source involved in the preparation of the euro zone finance ministers' meeting, called the Eurogroup.
"It is going to be a stock-taking meeting on the current state of discussions," a second source said. "Decisions should not be expected until next Monday."
Euro zone finance ministers will meet again next Monday in Luxembourg, in an attempt to have a solution ready for further financing for Greece in time for approval by a European Union leaders' summit on June 23-24.
The ministers will discuss Germany's idea that private investors would be offered a chance to exchange their Greek bonds for new ones, that would be seven years longer in maturity.
Some other euro zone countries back this proposal but the European Central Bank is against it, believing it would entail rating downgrades from credit rating agencies, which could be as deep as to a "default" level.
Berlin believes such a solution would trigger downgrades, but not to default level.
The European Commission supports a different idea -- to ask banks voluntarily to roll over their existing Greek bond portfolios as they mature and then replace the maturing bonds with longer maturity paper.
Such a solution would honor the terms of existing bond contracts until they expire, making it harder for credit rating agencies to declare this solution a default.
Olli Rehn, the European Commissioner For Economic And Monetary Affairs told Sueddeutsche Zeitung an accord was not as far off as some might think.
"We are preparing an agreement on the basis of the 'Vienna Initiative', whereby banks keep their bonds longer and in fact voluntarily," he said, referring to a debt rollover in which banks that hold Greek bonds are encouraged to buy more as their holdings mature.
"We are prepared to look at a solution, which is based on a voluntary extension of bond maturities and which under no circumstances leads to a credit default," Rehn said.
Euro zone policymakers are desperate to avoid a default because it would make Greek bonds unacceptable as collateral in European Central Bank liquidity operations, raising the specter of a Greek banking sector collapse.
French and German banks back private sector involvement in the new financing package for Greece.
Together with some 30 billion euros expected from Greek privatization revenues, the overall rescue package could total up to 120 billion euros between 2011 and 2014.
This would be on top of the remainder of the loans from the first financing package agreed last year, of which 45 billion euros will remain undisbursed once the EU and the International Monetary Fund pay out the next, 12 billion tranche of loans in early July.
The euro zone meeting will be followed by a dinner of all the EU's 27 finance ministers, who will discuss changes to the 27-nation bloc's budget rules to prevent debt crises in the future.
The changes include more and swifter sanctions for countries running budget deficits and debt above the EU limits of 3 percent and 60 percent of GDP respectively.
The key point of discussions will be on the method of voting when the ministers discuss whether a country breaking the rules has taken corrective action.
(Reporting by Jan Strupczewski, editing by Rex Merrifield)