The European Central Bank says it is not increasing its emergency funding for Greek banks, amid fears that Greece may default on its debts on Tuesday.
The decision not to raise the cap on aid to Greece increases the likelihood of bank closures and restrictions on cash withdrawals, analysts say.
That in turn could eventually result in Greece leaving the euro.
The ECB said that it stood ready to review the decision and would work closely with the Bank of Greece.
The current ceiling for the ECB's emergency funding - Emergency Liquidity Assistance (ELA) - is €89bn (£63bn). It is not clear if all that money has been disbursed.
Those funds are used by banks to provide cash to depositors who want their money back.
The ELA decision may mean Greek banks do not open on Monday to prevent a run ahead of the imposition of capital controls, BBC economics editor Robert Peston says,
That would be a significant step towards Greece leaving the euro, though it would not make it inevitable, he says.
Queues continued to build outside some ATMs in Athens on Sunday, but cash was readily available at many of them.
Greece's finance ministry said the country's financial stability council would meet later on Sunday to consider the situation.
Asked if capital controls were now inevitable, Greek Finance Minister Yanis Varoufakis told the BBC: "This is a matter that we'll have to work overnight on, with the appropriate authorities both here in Greece and in Frankfurt."
The ECB's move comes at the end of a turbulent week of negotiations between Greece and its creditors.
Weekend of turmoil
- Friday evening: Greek prime minister calls referendum on terms of new bailout deal, asks for extension of existing bailout
- Saturday afternoon: Eurozone finance ministers refuse to extend existing bailout beyond Tuesday
- Saturday evening: Greek parliament backs referendum for 5 July
- Sunday afternoon: ECB says it is not increasing emergency assistance to Greece
On Saturday, eurozone finance ministers refused a Greek request to extend Greece's current bailout beyond 30 June, the same day that Greece has to make a payment of €1.6bn to the International Monetary Fund (IMF).
Without new funds, Greece risks defaulting.
Greece and its creditors had been locked in talks over fresh bailout money on Friday when the Greek government broke off discussions to call a surprise referendum for 5 July over the terms it was being offered.
It described the terms as "not viable", and asked for an extension of its current deal until after the vote was completed.
Analysis: Andrew Walker, BBC economics correspondent
Some sort of restrictions on withdrawals, perhaps a bank holiday, look very likely for Monday.
The ECB was prepared to risk this consequence of their decision to restrict the ELA because the failure of the bailout talks cast new doubt on the viability of Greek banks - some of their assets depend on the government being able to meet its financial commitments.
And it is a fundamental principle of central banking that while you do lend to banks that are temporary difficulty, you only do so if they are solvent.
Mr Varoufakis said the prospect of Greece's creditors ending their financial support was "appalling".
It would mean "that Europe has failed in its duty to preserve in parallel a democratic process and a monetary union", he said.
Many Greeks still believe the banking system will survive the current uncertainty, the BBC's Joe Miller reports from Athens.
"We're going to be just fine," said a pensioner who voted for Syriza, the left-wing party of Greek prime minister Alexis Tsipras. "I trust him," she added, saying she would vote against accepting the creditor's proposals in the upcoming referendum.
In the queue at the cashpoint: Anna Holligan, BBC News, Athens
People standing in the queues for the cashpoint are heatedly debating politics. That's all you hear them talking about. The disagreements climax, the talking turns into shouting and now insults are being thrown - they are calling each other ignorant, naive, foolish and suicidal.
All discussions seem to lead to the question: "So who did you vote for in January?" Those who supported Syriza defend it by arguing that at least someone is standing up for the country, but they are attacked by others as responsible for the bank run.
"They took too much risk" is the feeling among many people.
Obviously, they are also discussing the referendum. Older people realise that a Yes at the referendum means that pension schemes will need to be reformed.
I even saw two people depositing money, claiming this was the conscientious thing to do. One of them tried to convince others not to withdraw their cash.