An attempt to privatise the Hellenic Postbank flopped on Friday without any interest in bidding, a banking source said, adding that the bank would probably have to be broken up.
"Barring a last-minute development, (Hellenic Postbank) will have to be broken up" to attract interest, the source told AFP.
The Bank of Greece is expected to make an official announcement later in the day.
The top three Greek banks -- National Bank, Alpha Bank and Eurobank -- and smaller Attica Bank had expressed varying degrees of interest in the sale last week.A minor but well-capitalised lender, Hellenic Postbank took a serious blow last year from a restructuring of Greek sovereign debt.
Its shares were suspended in August after a run by shareholders following statements from the finance minister that the bank had become "unsustainable".
The Greek state has a 34-percent direct stake in the postal bank and controls another 10 percent through its majority ownership of the Greek postal service, which is also up for sale.
A breakup solution was used at the end of July for ailing state lender ATEBank, the healthy part of which was absorbed by Greece's fourth-biggest bank, Piraeus Bank.