By Gabriele Steinhauser in Brussels and Andrea Thomas in Berlin 

Greece faces its European creditors at an impromptu summit of eurozone leaders Tuesday as both sides are scrambling to find a way to keep the country afloat after its emphatic rejection of tough bailout terms.

In a sign that Greek Prime Minister Alexis Tsipras is now giving priority to a rescue deal, the premier ousted his finance chief Yanis Varoufakis, whose confrontational style had alienated many other European finance ministers in negotiations, and replaced him with a less combative official, Euclid Tsakalotos.

Time is fast running out of time to reach a financing agreement before Greece's banks--already closed for most business for the past week--bleed dry of cash, threatening to plunge Greece's battered economy into even-deeper meltdown.

The head of the country's banking association said that banks would remain closed through Wednesday to stem the flow of money that Greeks had been pulling out of the banks; few expect them to open any time soon after that.

Without a new bailout deal, Greece also faces a default July 20 on a EUR3.5 billion ($3.9 billion) bond repayment to the European Central Bank. If the country is unable to finance its government or its banks with Europe's help, it could soon be forced to print its own national currency.

European officials sent conflicting messages on Monday about how they would respond to Greece's referendum result on Sunday, when more than 61% of Greek voters rejected creditors' conditions of further austerity for aid. France led calls to redouble efforts to reach an agreement with Athens and suggested that debt relief was a possibility, while Germany signaled that it doesn't want to offer Greece more lenient bailout terms.

"We have shown a lot of solidarity with Greece and the last offer [of bailout aid] was also a generous offer," Ms. Merkel said after meeting French President François Hollande in Paris to discuss the now-perilous impasse between Athens and its creditors.

Bolstered by its thumping referendum victory, Mr. Tsipras's government sought a consensus with opposition parties for a new bailout proposal to present to Greece's creditors, although it offered no economic-policy specifics on Monday.

Yet common ground between Greece and other eurozone governments is more elusive than ever, increasing doubts about Greece's ability to secure urgently needed financing for its state and banks and stay in the euro.

Greek voters' resounding rejection of the additional pension cuts and tax increases that creditors' demanded leaves few ways for Athens and Europe to secure a fresh bailout deal--unless Germany and others in Europe drop their resistance to restructuring Greek debt, many economists say.

The International Monetary Fund is insisting on "comprehensive" debt relief as an essential component of any further Greek bailout, a demand that puts it in line with the government in Athens. Less-intense austerity in Greece is only sustainable if the country's debt burden is reduced, the IMF has argued.

The White House also renewed its calls for Athens to restructure its economy and for Europe to offer the beleaguered county hope by offering better terms on its debt. Keeping Greece in the eurozone "will require both a package of financing and reforms that will allow Greece to achieve, or at least be on a path towards, some debt sustainability," White House press secretary Josh Earnest said Monday. "But it is also important for Greece to implement the kinds of reforms and to keep the commitments that they've made previously."

Debt relief might allow Mr. Tsipras to claim a political victory and sign up to otherwise politically difficult changes to Greece's pension and tax system, analysts say. But Germany and other eurozone governments have so far insisted that Greece complete its economic overhauls before any debt relief is considered.

Greece's debt burden, and the measures that Athens must offer in return for fresh loans, are expected to be at the center of discussions when eurozone leaders gather in Brussels on Tuesday. Ms. Merkel and Mr. Hollande said it was now up to Greece to submit proposals on what economic policies it is prepared to follow in return for renewed support from the other 18 eurozone countries.

"Time is pressing here and it is important to us that such proposals must come this week so we can resolve the situation," Ms. Merkel said.

The ECB on Monday increased its collateral requirements for Greek banks borrowing emergency funds from Greece's central bank--a warning shot that has no immediate consequences for the country's lenders but signals that the ECB wants a political agreement soon.

Such a deal could prove expensive for other European governments, both financially and politically. German officials on Monday reiterated their opposition to forgiving any of Greece's obligations.

"We are not discussing a haircut" on Greek debt, Martin Jäger, spokesman for German Finance Minister Wolfgang Schäuble, said Monday.

France's finance minister was more sympathetic. "I've always said talking about debt is not a taboo," said Michel Sapin. "The burden of debt in the coming months and years is too high for Greece to be able to pick itself up again."

Any deal on the debt would have to come as part of a broader agreement on more bailout aid. In coming years Greece has to repay billions of euros of loans from the IMF, which wouldn't be included in any restructuring.

Germany's vice chancellor and leader of the Social Democrats, Ms. Merkel's junior coalition partner, struck a pessimistic note on the likelihood of reaching a deal with Greece quickly.

"The ultimate insolvency of the country seems to be imminent," said Sigmar Gabriel, who is also Germany's economy minister. "We all know that it is pretty much impossible" for Greece to pay back its debt in the near- and long-term, he said.

There are alternatives to an outright reduction in the face value of the eurozone's rescue loans, such as cutting borrowing rates and extending debt maturities. That is what was being considered by the eurozone in 2012, when Greece last came close to a euro exit and the bloc's finance chiefs pledged to take steps to bring the country's debt below 110% of GDP by 2022. The IMF says that goal can no longer be reached without a haircut, however.

The expiration of Greece's eurozone bailout last week means that "this offer is not on the table anymore now," said Valdis Dombrovskis, a top European Commission economic official. Still, he added: "eurozone countries are ready to look at the Greek debt."

William Horobin, Ian Talley, Nektaria Stamouli and Brian Blackstone contributed to this article.

Write to Andrea Thomas at andrea.thomas@wsj.com