Austria's Finance Minister says a Greek exit from the euro appears almost inevitable.
Athens has called a 5 July referendum so that Greek voters can decide on whether to accept new, tougher bailout terms that the government itself opposes.
Austria's finance minister Dr Hans Jorg Schelling said the consequences for the euro countries were not nearly as bad as for Greece.
And he said it was clear that one country could under no circumstances blackmail the European Commission and the euro countries.
Meanwhile, the European Central Bank is expected to end emergency lending to Greece's banks on Sunday.
The country's banks depend on the ECB's Emergency Liquidity Assistance (ELA). Its governing council is meeting later.
Greece will probably have to "announce a bank holiday on Monday, pending the introduction of capital controls", a source told the BBC.
The bailout for heavily indebted Greece expires on Tuesday and talks have broken down.
Greek banks would find themselves in serious straits as soon as Monday if the ECB went ahead and cut the lifeline, the BBC's economics editor said.
Capital controls are restrictions on how much customers can withdraw from banks. Until now, the Greek government has signalled that it does not want to impose such controls.
In recent weeks, Greeks have withdrawn billions of euros from banks, and long queues formed at cashpoints on Saturday, amid fears that banks would not open on Monday.
The ECB has been sending emergency funds on a daily basis to the Greek central bank, which then allocates it to the high-street banks.
Greek MPs back bailout referendum
Greece's parliament has approved controversial plans to hold a referendum on international creditors' terms for a bailout.
Eurozone finance ministers had earlier rejected a Greek request to extend a bailout programme beyond 30 June.
A Eurogroup statement said Greece had broken off negotiations over a new bailout deal "unilaterally".
Greek Prime Minister Alexis Tsipras called a surprise referendum for 5 July over the terms of any new deal, which has just been authorised in the Greek parliament.
Greece has to pay €1.6 billion ($NZ2.3 billion) to the IMF on Tuesday. Without new funds, there are fears Greece may leave the euro and its economy may collapse.
French Finance Minister Michel Sapin stressed after the Eurogroup talks that all of its members wanted Greece to remain in the eurozone.
"This is not a Greek exit from the euro zone," he told reporters. "The 18 countries, apart from Greece, all said clearly that Greece was in the euro and should remain in the euro whatever the difficulties of the moment."
Greek Finance Minister Yanis Varoufakis said the Eurogroup's refusal to extend the bailout could permanently damage the credibility of the group.
He said that what had been proposed to Greece "did not contain any plan for giving, instilling hope in investors, both Greek and non-Greek, in consumers, in depositors".
But Mr Varoufakis said Greece would still try to secure a bailout deal that could then be put to the Greek people in a referendum.
"In these crucial moments, the Greek government is fighting for there to be a last minute deal by Tuesday," he said.
Mr Varoufakis said his government had asked for an extension of "a few days, a couple of weeks", whereas Mr Dijsselbloem said an extension of one month had been requested.
"How does the Greek government think that it will survive and deal with its problems in that period? I do not know," he said.
Greece owes roughly €340 billion, mostly to its eurozone partners. Because it can no longer borrow from the international money markets, it is dependent on the eurozone and IMF to keep its banks functioning.
-BBC / Reuters