The "good" assets of the banks will be transferred to Intesa Sanpaolo SA (ISP.MI) for the token price of EUR1.
Italy's government is bailing out two banks in the Venice region at a cost of 5.2bn euros (£4.6bn; $5.8bn).
The Italian banking fund Atlante invested?3.5 billion ($3.9 billion) in Veneto Banca and Banca Popolare di Vicenza in 2016.
MILAN, June 26 European shares got off to a firm start to the week on Monday as banks rallied after Italy reached a deal on two failed regional banks and consumer bellwether Nestle hit a record high after becoming the next target of activist investor Third Point.
The Italian plan instead takes advantage of an exception to European Union bank rules that allows the use of routine insolvency proceedings with banks not considered systemically important, allowing the process to be handled by the member state. To do that, however, the government has potentially used up the whole €20-billion it had set aside to help ailing banks. They will now face insolvency proceedings in Italy.
Italy's Economy and Finance Ministry said the government would meet over the weekend to "adopt the necessary measures to keep the banks fully operative, protecting all account holders, depositors and senior creditors".
Rome will provide a further "guarantee" of 400 million euros, Padoan said, with the remaining cash going to cover a huge hole due to bad loans.
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 almost 350 billion euros of soured debts - a third of the euro zone's total.
Pressure on Rome to find a solution for the two Veneto lenders had increased since Spain's Banco Popular POP.MC was rescued by Santander this month in a deal orchestrated by European authorities. At the end of past year, it had to promise billions of euros to rescue Monte dei Paschi, the world's oldest operating bank.