Greece said it may impose capital controls and keep its banks shut on Monday after creditors refused to extend the country's bailout and savers queued to withdraw cash, taking Athens' standoff with the European Union and the International Monetary Fund to a dangerous new level.

Greece's banks, kept afloat by emergency central bank funding, are on the front line as Athens moves towards defaulting on a 1.6 billion euros payment due to the IMF on Tuesday.

The European Central Bank said it would not raise the level of emergency funding, adding to the pressure on Greece's banks which have been surviving for the past few weeks on frequent incremental increases to the funding lifeline

Amid political drama in Greece, where a clear majority wants to remain inside the euro, the next few days present a major challenge to the integrity of a 16-year-old currency bloc.

"This is a matter that we'll have to work overnight on with the appropriate authorities both here in Greece and in Frankfurt," Greek Finance Minister Yanis Varoufakis said of bank closures and capital controls. He was speaking to BBC radio.

The finance ministry later issued a statement saying capital controls were not the government's preference and were not consistent with monetary union.

Greece's left-wing Syriza government had been negotiating a deal to release funding in time for its IMF payment. Then suddenly, in the early hours of Saturday, Prime Minster Alexis Tspiras asked for extra time to enable Greeks to vote in a referendum on the terms of the deal.

Creditors flatly turned down this request, leaving little option for Greece but to default, piling further pressure on the banking system.

Long lines formed outside many ATMs on Sunday, including some of 40 to 50 people outside some in central Athens. The German foreign ministry issued a travel warning advising tourists heading to Greece to take plenty of cash to avoid possible problems with local banks.

The Bank of Greece said it was making "huge efforts" to ensure the machines remained stocked.

The ECB said in a statement on Sunday it was keeping its emergency liquidity at current levels but was monitoring the situation and stood ready "to reconsider its decision."

There is growing opposition in some European countries to extending the ECB's funding line.

In economic powerhouse Germany, other southern states that have suffered austerity in return for EU cash and poor eastern countries with living standards much lower than Greece's, many voters and politicians have run out of patience.

If Greece were to leave the euro zone, the ECB's funding line, which is a form of overdraft with euro zone's central bank system, would fall to the bloc's other members to pay.

The head of Germany's Bundesbank has attacked the use of the ECB's funding line and speaking on German television on Saturday, Finance Minister Wolfgang Schaeuble put a question mark over the solvency of Greek banks - a key condition to qualify to receive such finance.

"The ECB has always said that as long as Greek banks are solvent, then emergency loans, the ELA, can be granted," he said.

"And now there is naturally a new situation that because of the developments the liquidity and solvency of Greek banks, or some Greek banks, could be in doubt."

German Chancellor Angela Merkel has invited leaders of all the major German parties to a meeting in Berlin on Monday to discuss the crisis.

DEAL STILL POSSIBLE

The 18 other countries sharing the euro countries have blamed Greece for breaking off negotiations and pledged to do whatever it takes to stabilize the common currency area.

But some officials said there was still time to return to the negotiating table.

French Prime Minister Manuel Valls on Sunday urged the Greeks to continue talks, warning that the country's membership of the euro was at stake.

"A deal is still possible, I invite the Greek government to come back to the negotiation table," Valls told Europe 1, Le Monde and iTELE in a joint interview. "I cannot resign myself to Greece leaving the euro zone ... We must find a solution."

International Monetary Fund boss Christine Lagarde told the BBC that there was still time for the Greek government to change course.

She also said that if the July 5 vote produced "a resounding yes" to remain in the euro and fix the Greek economy then the creditors would be willing to make an effort.

Pro-European Greek opposition parties have united in condemning the decision to call the referendum. There was rising speculation that Tsipras' left-wing government may have to resign if voters back the bailout in the referendum.

The offer from creditors requires Greece to cut pensions and raise taxes in ways that Tsipras has long argued would deepen one of the worst economic crises of modern times in a country where a quarter of the workforce is already unemployed.

European Council President Donald Tusk said on Sunday he was in contact with all the governments of the euro zone to ensure Greece remained in the single currency.