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.
Privatisations are a central component of reforms which Greece has to enact as part of its debt bailout conditions, but the programme is running late and has had to be scaled down.
"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.
But according to reports, National Bank and Eurobank are mainly interested in an ongoing merger process while Alpha Bank was put off by the prospect of having to guarantee Hellenic Postbank's workforce of over 3,000.
National Bank said on Friday that on Monday the Hellenic capital market commission had approved its public offer to Eurobank shareholders for a share exchange.
The stock swap will begin later on Friday and be concluded on February 15, National Bank said, and the results will be announced by February 18.
Eurobank shareholders will receive 58 shares in the new group for every 100 old shares.
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.
Under the terms of Greece's EU-IMF loan bailout, the government plans to raise 2.5 billion euros ($3.3 billion) in asset sales in 2013.
Other companies to be sold in coming months include gas distributor DEPA and gaming monopoly OPAP.
Greece was originally supposed to raise 50 billion euros from asset sales by 2015.
This was later scaled down to 19 billion, and in October the government said it planned to raise just 9.5 billion euros by 2016.
In contrast, the European Commission expects 8.5 billion euros in sales by 2016, and 22 billion euros by 2020.