The European Union turned up the heat on bickering Greek politicians Sunday to agree on a crisis coalition, demanding progress toward backing an international bailout deal in the next 24 hours.

In a sign Greece's political deadlock may be easing under EU pressure, a senior socialist said Prime Minister George Papandreou made clear he would resign once a coalition deal was done, possibly as soon as Sunday night.

With Eurozone finance ministers due to meet Monday, senior socialist lawmaker Telemachos Hitiris said, Everything must be done within the day, otherwise tomorrow it will be hell.

Earlier, the conservative opposition offered to cooperate in forming a national unity government that would push the Eurozone bailout through parliament, provided Papandreou stood aside after two years at the helm as Greece slid toward bankruptcy.

European Economic and Monetary Affairs Commissioner Olli Rehn demanded a Greek unity government to restore confidence, which has been shaken by doubts that Athens would commit itself to the 130 billion euro bailout package. Rehn wanted progress by the time Eurozone finance ministers -- including Greek Deputy Prime Minister Evangelos Venizelos -- meet Monday night.

We have called for a national unity government and remain persuaded that it is the convincing way of restoring confidence and meeting the commitments, he told Reuters.

We need a convincing report on this by Finance Minister Venizelos tomorrow in the Eurogroup.

A Breach of Confidence

Greece had breached confidence with its Eurozone partners last week and put itself on a path toward leaving the common currency, Rehn said in a telephone interview.

Papandreou provoked uproar last Monday by announcing a referendum on the bailout, which demands yet more austerity to be imposed on the long-suffering Greek population, plunging his country into political as well as economic crisis.

Under heavy pressure from home and abroad, Papandreou ditched the referendum and narrowly survived a confidence vote, but only after promising to seek a new broad-based coalition. Rehn said Greece now appeared to be pulling back from the brink.

Early on Sunday, the left and right had seemed far apart and any coalition deal between Papandreou's PASOK socialists and the conservative New Democracy a distant prospect.

But the EU pressure appeared to start working. With the cabinet due to meet later Sunday, Hitiris said explicitly what Papandreou had signaled in parliament Friday.

Papandreou will resign once the new prime minister is selected, he told state NET TV. We only have to wait for the prime minister's announcements in the cabinet.

Torturing the Greek People

In the hunt for a national consensus, President Karolos Papoulias met the conservative opposition leader Sunday.

This uncertainty that is torturing the Greek people must end. We must find a solution, Papoulias said before starting closed-door talks with New Democracy leader Antonis Samaras.

Samaras said he was ready for compromise, but only with a new prime minister. I am determined to help. Provided that Papandreou resigns, everything will take its course, he said after the talks.

The bailout deal aims to save Greece from bankruptcy and to prevent its problems tipping much bigger Eurozone economies such as Italy and Spain into full-blown crisis.

Venizelos has already received one lecture when German Chancellor Angela Merkel and French President Nicolas Sarkozy summoned him with Papandreou before a Group of 20 meeting last week.

The leaders made clear that Greece would receive not one cent more in European aid until it had signed up to the latest bailout, the second package since Athens had to go cap in hand to the European Union and International Monetary Fund in May last year.

Greeks have fought the pay and pension cuts, combined with higher taxes, demanded by the international lenders with a series of strikes and protests, some violent.

But the state is due to run out of money in December, when it has big debt repayments to meet, and by then needs the sixth installment of its existing EU/IMF bailout package.

A senior New Democracy official said Samaras was willing to negotiate a government of politicians, rather than one comprising technocrats, if Papandreou stepped aside. Party sources said he was under pressure from his own lawmakers, who feared the consequences if they were left out of a coalition.

Having demanded snap elections, he might be willing to discuss a date later than his preferred early December vote.

The source made no mention of who might lead the government, but New Democracy is unlikely to accept any top member of the ruling PASOK party such as Venizelos.

Only a week ago, the bailout deal seemed in the bag, but then Papandreou dropped the referendum bombshell, prompting widespread fear that Greece might be forced out of the euro and have to go it alone with a revived national currency.

Europeans don't trust us anymore; they will throw us out, said Tassos Pagonis, a 48-year-old taxi driver in Athens. I hope we don't return to the drachma.

(Additional reporting by Lefteris Papadimas, Rene Maltezou and Harry Papachristou; Writing by David Stamp; Editing by Kevin Liffey)