Intesa Sanpaolo, Italy's best-capitalized large bank, said last week it was open to purchasing the rump of the good assets for one euro on condition Italy's government passed a decree agreeing to shoulder the cost of winding down the two banks.
Setting tough conditions for the deal, Intesa CEO Carlo Messina has insisted that his bank's capital ratios and dividend policy would not be affected by the transaction.
"Without Intesa Sanpaolo's offer - the only significant one submitted at the auction held by the government - the crisis of the two banks would have had a serious impact on the whole Italian banking system," Messina said in a statement on Sunday.
The European Central Bank, which supervises the two lenders based in the country's rich north-eastern Veneto region, had declared on Friday that they were "failing or likely to fail", setting in motion the process that led to them being wound down.
After months of tense negotiations between Rome and EU regulators, Italy has been allowed to use national insolvency procedures rather than EU rules designed to prevent the use of state money to deal with bank crises.
Those EU rules could have imposed losses on senior bondholders and large depositors, a politically unpalatable prospect ahead of national elections next year given that Italian households hold a large chunk of bonds issued by banks.
Some European officials have voiced exasperation at the way Italy has dealt with a string of trouble spots in its banking industry, which is weighed down by nearly 350 billion euros of soured debts -- a third of the euro zone's total.
The country's fourth-biggest bank, Monte dei Paschi di Siena, is being bailed out by the state to cover a capital shortfall of 8.8 billion euros.
Four other smaller banks were wound down in 2015. The future of the Veneto banks had hung in the balance over the past two years since the ECB uncovered a capital hole caused by a spike in bad loans and a mis-selling scandal whereby the banks' customers were sold shares in exchange for loans.
The dragged-out negotiations between Rome, Frankfurt and Brussels and their outcome on Sunday have raised serious questions about the effectiveness of banking supervision and the credibility of Europe's own rules for dealing with bank crises.
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