By Gabriele Steinhauser And Matthew Dalton 

BRUSSELS--Eurozone leaders set Greece a Sunday deadline to come up with new and even-tougher economic measures if the country wants to avoid defaulting on the European Central Bank and crashing out of the currency union.

As a sweetener for such a deal, leaders raised the possibility of some short-term financing to help Athens make a July 20 payment and--most important for Greek Prime Minister Alexis Tsipras--action down the road to relieve Greece's crushing debt burden.

Obstacles to an agreement that keeps Greece in the eurozone remain high, however. Most notably, the policy overhauls and budget cuts demanded go beyond those that were resoundingly rejected by Greek voters in a referendum last weekend.

German Chancellor Angela Merkel said after Tuesday's emergency summit of eurozone leaders that it is up to Greece to act. "Of course, at the very end, one will have to discuss how debt sustainability can be recreated but not by saying first 'How do we close the gap?' but "What can Greece do?' " she said.

She added that Mario Draghi, the ECB president, made clear to leaders at the summit that Sunday would be "the right moment to take decisions" for Greece to avoid a meltdown of its banking system.

Mr. Tsipras said that the negotiation process would be fast. "I believe there will be a solution by the end of the week at the latest," he said.

Leaders said that the broad outlines of such an agreement needed to be nailed down at a new summit on Sunday of the leaders from the entire European Union, including nine countries that aren't in the eurozone.

Without some new cash in the coming weeks, Greece won't be able to make the EUR3.5 billion ($3.8 billion) bond payment to the ECB, leading to the country's second default in less than a month.

Such a nonpayment could push the ECB to cut emergency lending to Greek banks--a move that would send the country's financial system into meltdown and force the government to print its own money to recapitalize them.

Greek banks have been shut for more than a week, after the ECB put a limit on the emergency loans they had been drawing to buffer growing deposit outflows. Cash withdrawals from ATMs have been limited to EUR60 a day and depositors are unable to transfer money abroad.

Some short-term funding and the prospect of a longer-term bailout deal from the rest of the eurozone could allow Greece--and the ECB--to let banks reopen and normal economic activity in the country to resume.

Getting there, however, would require a formidable turnaround by Mr. Tsipras and his government.

In addition to the pension cuts and tax increases, rejected in the referendum, a new rescue package to keep the government afloat for two to three years would require overhaul measures that had dropped out of negotiations in recent months.

These, Ms. Merkel said, include changes to labor laws to make it easier to fire workers, changes to product markets and the privatization of state assets.

"This isn't about a program from today, or from yesterday or from 10 days ago, for a program until November...this program is, according to the now-withdrawn Greek request, meant to be two years long, so it is a multiyear program," Ms. Merkel said.

"That such a multiyear program has to entail more commitments is in itself clear," she added.

Only once that is clear, leaders would look at ways to make Greece's debt sustainable again, Ms. Merkel said, adding that any debt relief wouldn't include cuts to the nominal value of rescue loans.

But talks on details of a final plan could drag on for weeks.

"The short-term deal is only possible in a longer-term deal," French President François Hollande said at his postsummit news conference. "And the short-term deal has conditionality."

A senior French official pushed the idea that Greece could get emergency financing by passing just the budget measures of a potential agreement through the Greek parliament, and passing non-budgetary, more structural changes later. But Germany may not accept that.

Emboldened by Sunday's referendum, Mr. Tsipras came to Brussels to present his eurozone counterparts a proposal for interim financing until the end of the month, a senior Greek government official said.

That would give Greece and its creditors time to work out a longer-term financing deal and avoid defaulting on the ECB.

Mr. Tsipras was hoping to use the leverage from his referendum victory to win enough of a political nudge from the summit that would allow the ECB to keep its lifeline in place.

He raised the idea of interim financing with Ms. Merkel, Mr. Hollande and Jean-Claude Juncker, head of the European Union's executive Commission, ahead of the summit, the Greek official said.

Whether the new reform commitments from Mr. Tsipras were credible "I can't say today." Ms. Merkel said later.

At a meeting of finance ministers earlier Tuesday, Greece's new finance chief, Euclid Tsakalotos, read out proposals the government had presented last week, three European officials said.

But the new finance minister's performance raised some hopes among ministers that things had changed. "The minister Tsakalotos came for the first time in this meeting and he adopted clearly a constructive, moderate listening mode, which I think was appreciated by all the colleagues," Pierre Moscovici, the EU's economics commissioner said.

The International Monetary Fund, which Greece technically defaulted on last month, has said "comprehensive" debt relief should be included in any further bailout. But divisions still run deep in Europe.

Leaders warned Tuesday that failure to strike a deal could catapult Greece outside the common currency.

"We have a Grexit scenario prepared in detail," said Mr. Juncker.

Nektaria Stamouli and Todd Buell contributed to this article

Write to Gabriele Steinhauser at gabriele.steinhauser@wsj.com and Matthew Dalton at Matthew.Dalton@wsj.com