European leaders faced a dilemma Monday about whether to back down in the face of a resounding Greek rejection of their austerity demands, amid signs that Greece was preparing a more conciliatory bid for financial rescue just days before its banks face collapse.

All eyes were on a European Union conclave on Tuesday that promises to be decisive for Greece, whose banks risk running run out of cash within days. The pressure intensified after the European Central Bank decided Monday not to expand its life support for the Greek lenders, raising the prospect that they could be out of money by Wednesday absent an ECB reversal. That could force Greece off the euro, the common currency of 19 of the E.U.’s 28 member states.

Europe’s decisions about how to respond to Sunday’s Greek referendum, which rejected Europe’s exacting bailout terms, will determine far more than the fate of the flailing Mediterranean nation. Other euro-zone countries with shaky finances, such as Spain and Portugal, are watching to see whether Europe backs down from its tough economic demands in the face of domestic rebellions. That would spark new pressures in those nations. And if Greece is kicked off the euro, the ensuing turmoil might inspire efforts in other nations to quit the currency club.

E.U. leaders said Monday that they were awaiting Greece’s new bailout proposals, hinting at the possibility of a compromise despite vows last week that there would be no further discussion if Greeks rejected Europe’s terms,

Greeks overwhelmingly voted against a strict bailout package proposed by the E.U. Here is how that affects Greece and the United States. (The Washington Post)

“We want to keep this one currency, and therefore shared responsibility is necessary on all sides, just like shared solidarity,” German Chancellor Angela Merkel told reporters in Paris, where she was meeting with French President François Hollande for emergency talks about Greece. Merkel also spoke by phone with Greek Prime Minister Alexis Tsipras while he was at a marathon meeting with the heads of Greece’s main political parties.

Although Tsipras won a resounding mandate Sunday from Greek voters to reject Europe’s bailout demands, it remained far from clear Monday whether he could convert that victory into a win at the negotiating table.

The ECB decision is “putting pressure on all sides,” said a senior Greek banking official, who spoke on the condition of anonymity to clarify internal discussions. “We go day-by-day until a resolution is found.”

The official said that Greek banks had been provided with enough money to last until Wednesday evening. The ECB plans to revisit its decision that day after it knows the outcome of the Tuesday summit of euro-zone leaders, the official said.

The ECB left unspecified the timing of the extension of its $98 billion credit line, but it increased the pressure on Greek banks by raising the amount of assets that the banks need to muster as collateral to qualify for the emergency assistance.

That move could eventually lead to a bank failure, especially given the already tenuous positions of the Greek banks. But an ECB official, speaking under ground rules of anonymity, said Monday that the central bank did not expect the collateral provision to immediately lead to such a failure.

European leaders — in control of the funds that have kept Greece afloat — gave no immediate indication that they would rush to make a deal. Each day, they gain further leverage over Greece through its financial sector. Already, the banks have been shuttered for a week, and Greeks are limited to withdrawals of 60 euros (about $67) per day.

The dwindling cash is sucking the life out of everything from coffee shops to taxis, as anxious Greeks economize amid fears for the future. Greek leaders also banned transfers of money abroad, meaning that very little can be imported.

Printing plants are warning that they may run out of paper for newspapers by the end of the week. Butchers say stocks of imported meat are dwindling.

The rising shortages may speed a deal, some analysts said.

“Merkel does accept that Greece may leave the euro, but she doesn’t want to be blamed for it,” said Charles Grant, director of the London-based Center for European Reform, a policy institute. “She has to try and go the extra mile to make a compromise feasible.”

Greece’s leadership was readying a new bid for aid after the heads of the main opposition parties lent Tsipras their support following a meeting that lasted more than six hours. The unity stance was a vital boost that might pave the way for a deal.

The talks came after firebrand Finance Minister Yanis Varoufakis announced his resignation. That move also eased the path toward compromise by removing a man whose unbending style and angry denunciations of European economic “terrorism” earned him the enmity of nearly all the country’s creditors.

Varoufakis was one of the architects of Greece’s approach to the negotiations, pushing uncompromising stances that left little room for the give-and-take common when deals are being hammered out.

Named as his replacement was a mild-mannered Oxford-trained economist, Euclid Tsakalotos. He has headed Greece’s negotiating team since April.

Tsakalotos vowed Monday to deliver relief to Greeks, who have suffered from five years of economic pain rivaling the Great Depression.

“People who are poor, middle-aged, unemployed — they said that they trust their government to find a viable solution,” he said after being sworn into office.

After his last appearance at the ministry as its head, Varoufakis, by contrast, rode off with his wife on a motorcycle, befitting his bad-boy image.

“I shall wear the creditors’ loathing with pride,” Varoufakis wrote in his resignation message.

The resounding victory Sunday for Greeks who said no to European austerity demands — 61­­ percent in all — gave a much-needed domestic boost to Greece’s battle-scarred leftist leaders.

Tsipras “clearly has come out of this referendum as the master of the game as far as the domestic politics are concerned,” said Loukas Tsoukalis, president of the Hellenic Foundation for European and Foreign Policy, an Athens-based research organization. “Whatever you think of him, nobody can challenge him.”

The triumph could only be temporary, since Greece owes billions of euros to its creditors on July 20 but has no money to pay them.

Despite Merkel’s conciliatory words Monday, her vice chancellor and economic affairs minister, Sigmar Gabriel, warned that Greece may soon need humanitarian aid He said Europe needs to begin preparing emergency medicine and food shipments in the event that negotiations break down.

Other E.U. leaders also warned against expecting a swift compromise.

“The outcome of the referendum certainly has made things more complicated,” said European Commission Vice President Valdis Dombrovskis. “There are going to be very few winners in this situation.”

Griff Witte and Ylan Q. Mui in Athens and Stephanie Kirchner in Berlin contributed to this report.

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