Greek savers may be gripped by a "great fear that could develop into panic" in the words of President Karolos Papoulias, but many Greeks shifted their money to safer havens in Britain, Switzerland, Germany and Nordic countries long ago.
Worries about a run on Greek banks has rattled Athens this week, after savers withdrew at least 700 million euros ($891 million) on Monday alone, according to minutes of Papoulias's comments to political leaders posted on the presidency's website.
It is not only Greeks who are worried about their savings. Data shows depositors have also taken flight from banks in Belgium, France and Italy.
And on Thursday, Spain's Bankia was reported to have seen more than 1 billion euros drained by its customers in the past week.
Greeks are afraid they could be hit by rapid devaluation if the country leaves the European single currency, while customers at Bankia have been rattled by the government's takeover of the recently floated bank on May 9 and growing uncertainty about the final cost of Spain's banking reforms.
In Greece, sources at two banks told Reuters that withdrawals on Tuesday had taken place at about the same rate as on Monday.
"The entire Greek banking system is in danger: the banks are now facing the worst of all outcomes, deposit flight," said Arnaud Poutier, deputy CEO of IG Markets France.
That flight started at least two years ago, as the debt crisis grew more serious.
Greece's banks have lost 72 billion euros in deposits since the start of 2010, or about 30 percent, according to data compiled by Thomson Reuters. Five of Greece's top banks saw 37 billion euros taken out last year, including 12 billion from EFG Eurobank and 8-9 billion apiece at National Bank of Greece, Piraeus and Alpha Bank.
In February, Evangelos Venizelos, finance minister at the time, said only 16 billion euros had gone abroad, with a third of that going to Britain.
Savers have shifted to property, gold and other banks, or stashed it privately.
In Greece, this slow-speed run on deposits has not caused panic. But that could quickly change if there is a sudden loss of confidence in the banks.
Savers lost faith in Britain's Northern Rock overnight in September 2008, queuing for hours in the days that followed to take out their cash, despite a guarantee safeguarding most deposits. The British government ended up nationalizing the bank.
"It (Greek withdrawals) is not a huge number in percentage terms, but it is still a very worrying story. But deposit flight has been going on for two years. What we are seeing in the eurozone is a slow-motion bank run," said Michael Riddell, fund manager at M&G International Sovereign Bond Fund.
Deposits shifted around Europe dramatically last year, analysis of data from more than 120 listed European banks show.
More than 120 billion euros was taken from two banks in Belgium alone, including an exodus of customer deposits from Dexia which had to be bailed out and restructured. KBC also saw a big outflow.
Some 90 billion euros was taken from France's banks, including around 30 billion each from Credit Agricole and BNP Paribas. French banks were hit last year by their heavy exposure to Greece and concerns about their liquidity that forced them to accelerate plans to shrink.
Worries the eurozone crisis would spread also saw about 30 billion euros in deposits leave Italian banks, although inflows to BBVA helped limit the net outflow from Spain.
Cash flooded into Britain; more than 140 billion euros was deposited in four big banks alone. The UK benefits from its position outside the eurozone and its Asia-focused banks HSBC and Standard Chartered are seen as particular safe-havens.
Other banks to see big inflows included Barclays, Germany's Deutsche Bank, Switzerland's Credit Suisse and UBS and Russia's Sberbank and VTB.
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