* Talks over two banks expected to end by end-June -source
* Banks could be wound down if no solution found -source
* Italian treasury source rules out any bank "resolution"
BRUSSELS, May 30 (Reuters) - Italy's Banca Popolare di Vicenza and Veneto Banca may be wound down if a deal cannot be reached next month on granting the banks state aid, a European Union official said.
"By the end of June we should know whether a precautionary recapitalisation is doable," an EU official close to the talks told Reuters on Tuesday.
If no deal between Rome and EU authorities is found by then, European regulators would prepare for their "resolution" under "bail-in" rules, where shareholders, bondholders and large depositors shoulder losses, the official added.
Rome resolved to set aside 20 billion euros ($22 billion) to help its weaker lenders in December after the country's fourth biggest bank, Monte dei Paschi di Siena, failed to raise money on the market to plug a capital shortfall.
Italy's banking industry has 350 billion euros of bad loans, a third of the euro zone's total, which accumulated during a deep recession and the government is now negotiating with the EU executive a state bailout for Monte dei Paschi and the two Veneto-based regional lenders.
This would be under new European rules which came into full force in January 2016 and seek to shield taxpayers from the cost of bailing out banks.
But while talks on a state rescue for Monte dei Paschi appear to be heading towards a positive conclusion, negotiations
over the two Veneto banks are more difficult.
Italy is seeking to use an exception to the EU rules and win approval for a precautionary recapitalisation of the banks that would reduce private investors' losses and hit only shareholders and junior bondholders.
Brussels, however, is demanding that private investors inject an extra 1 billion euros into the two Veneto lenders to help fill a capital shortfall of 6.4 billion euros before state aid can be used, sources have said.
The Italian government last week ruled out the possibility of a bail-in for the two banks, fearing this would hurt confidence in the wider banking sector.
With the prospect of early elections growing after former Prime Minister Matteo Renzi said at the weekend he favored a vote in the autumn, the government is likely to be even more opposed to a potentially vote-losing resolution.
But it is struggling to persuade Atlante, a fund financed by healthier Italian banks, to provide the additional capital to save the two banks, sources have said.
Atlante, hastily set up last year to rescue the two Veneto lenders after their attempt to raise capital on the market flopped, has already pumped in 3.4 billion euros. Responding to a formal request for more help, it said on Tuesday it had no intention of stumping up more money.. "It's going to be a long tug-of-war where someone will eventually have to give in," an Italian banking source said.
Popolare di Vicenza, Veneto Banca and the European Central Bank declined to comment, while an Italian treasury source said there was no risk of resolution.
"The only option on which we are working is the precautionary recapitalisation, there are no other options. The discussions take time, for Monte dei Paschi the negotiation started five months ago," the source said.
The European Commission has repeatedly said there was no deadline for negotiations to solve the Veneto banks' woes.
Asked whether talks should be concluded by the end of June, a spokeswoman for the EU Commission declined to comment, reiterating that "constructive contacts are ongoing."
Under European banking rules, it is up to the ECB - the euro zone's single banking supervisor - to determine when a bank is failing or likely to fail because it does not have enough capital, cash or if its liabilities exceed its assets.
However, the Single Resolution Board - the EU resolution authority - can also take such a decision.
In a bail in, healthy parts of a failing bank could be salvaged, but shareholders, bondholders and even depositors with more than 100,000 euros ($111,580) would be hit.
Such a procedure has not been implemented since the new EU rules for dealing with bank crises came into force. ($1 = 0.8962 euros) (additional reporting by Stefano Bernabei and Giuseppe Fonte in Rome, editing by Silvia Aloisi and Alexander Smith)