Greeks withdrew as much as 5 billion euros -- nearly 3 percent of total deposits -- after outgoing Prime Minister George Papandreou's shock call last week for a referendum on a euro zone bailout, said one banker, who declined to be named.
"Many people withdrew their money from banks on Thursday and Friday and money couriers had a hard time supplying banks with cash to satisfy the emergency demand," said another banking source, who declined to be named.
Greece's central bank governor made a rare public appeal for a new government to take charge and end the dithering and has warned that Greece's euro zone membership is at stake.
"The (political) uncertainty is hurting the economy and the banking system," Bank of Greece governor George Provopoulos told Reuters. "There must be a strong government that will work hard to ensure the country's future in the euro zone."
European leaders have warned Athens that the country must show it can get its act together to secure bailout money before the government runs out of money in December.
Resigned to political bickering and a growing sense of chaos, Greeks headed to the banks in droves.
Many fear the Greek bank system could collapse and a sharp loss in the value of their deposits if Greece leaves the euro and is forced to adopt a new, weaker currency.
"We got to the point where customers ordered amounts of up to 600,000 to 700,000 euros in cash to take home -- unbelievable," the first banker said. "This strains the system."
He said his bank had resorted to offering interest rates of as much as 7 percent on term deposits to convince Greeks to keep their money in their accounts.
One bank clerk was overheard saying to a co-worker at an Athens branch of National Bank, the country's biggest bank: "A lot of money was withdrawn on Friday."
Deposits have fallen more than 21 percent since January 2010, when Greece's debt crisis shifted into a higher gear, and the banks have become increasingly reliant on the European Central Bank for their liquidity needs.
Latest figures show banking deposits fell about 3 percent in September to 183.2 billion, and the first banker said an additional 8 billion euros are likely to have been withdrawn from the system in October.
The latest withdrawals were mainly from retail bank accounts after an initial wave of wealthy private banking clients took their money abroad last year, a Greek private banker said.