With Greece's creditors refusing to extend its bailout, attention has turned swiftly to preventing massive capital flight as worried Greek citizens pull cash from ATMs.
Fears that the banks may not open Monday have prompted the European Central Bank to meet but officials say it will be up to Greece to stem an outflow that has already reached dangerous levels.
"If there isn't capital controls by Tuesday at the latest, it's over," said a European source close to the negotiations.
"Greek banks are near liquidation and can no longer remain solvent. Once the banks fail, 'Grexit' will become irreversible," the source said.
Talks on Saturday collapsed with the Greek contingent leaving the remaining 18 eurozone ministers to consider the consequences of a default.
In a statement, the ministers appeared to urge Greece towards capital controls, saying that the expiry of the bailout "will require measures by the Greek authorities, with the technical assistance of the institutions, to safeguard the stability of the Greek financial system."
The withdrawals today were "exceptionally high," warned the influential German Finance Minister Wolfgang Schaeuble at the talks on Saturday.
Faced with this calamity, Natixis analysts Jesus Castillo and Alan Lemagnen said the Greek government could decide a bank holiday, an order that the nation's banks remain closed to avoid a run by customers.
Similar moves were made in 2013, when Cyprus imposed drastic limitations on cash withdrawals and money transfers abroad when its banks faced crisis -- in large part due to contagion from the crisis in Greece.
- 'Fight contagion' -
"We will do everything to fight against any possible danger of contagion," Schaeuble added.
If the scenario were repeated in Greece now, European leaders would act to prevent the same contagion that flowed from Greece in 2012 to other troubled eurozone members like Spain, Portugal and Ireland.
That disruption was caused by spooked investors shunning the bonds of vulnerable countries, sending their borrowing rates unsustainably high. At the same time, banks in healthier eurozone countries holding debt of weak eurozone nations were suddenly seen as a risk.
But risk of renewed contagion has been greatly reduced by firewalls built since 2010 by eurozone authorities, and the creation of a European banking union to police lenders and oversee a collective response to the crisis.
And in the meantime, most European banks have significantly wound down their exposure to Greece and other troubled eurozone members.
The ECB also now has tools unavailable to it in 2010. It is in the midst of successful quantitive easing programme injecting liquidity into the eurozone economy, and could easily step up its purchases of sovereign bonds if investors dump debt.
It could also deploy the thus far unused "outright monetary transactions" to purchase sovereign debt -- a plan it unveiled in 2012 to calm panicked markets, and which has recently cleared legal challenges from opponents.
Until Saturday's debacle in Brussels, the ECB bought time to allow discussions to continue, pumping cash into the teetering Greek financial system.
To achieve that, the Frankfurt-based central bank maintained its emergency liquidity funding to Greek banks to prevent their collapse -- and in doing so withstood heated opposition from Germany.
So far ECB chief Mario Draghi has refused to cut emergency funding for Greek banks, but that decision is expected to be soon reversed.
The central bank's governing council is set to hold an emergency meeting on Sunday, and a decision to end the lifeline is increasingly expected.