* EU told banks to raise 1 billion euros in private capital
* Pop Vicenza, Veneto Banca seek help from Atlante fund
* Atlante's investors unwilling to stump up more -sources
* Italy keen to avert "bail in" of investors in both banks (Adds sources, background)
By Valentina Za and Gianluca Semeraro
MILAN, May 26 (Reuters) - Banca Popolare di Vicenza and Veneto Banca have asked an investment fund backed by Italy's healthier banks to come to their rescue by helping to cover a capital shortfall, as demanded by the European Union.
Rome has sought EU approval for months to bail out the two Veneto-based lenders and bigger rival Monte dei Paschi di Siena under strict EU bank rules that curb state support.
Popolare di Vicenza and Veneto Banca have requested state aid to fill a capital gap of 6.4 billion euros ($7.2 billion), but the European Commission has said they must first find an additional 1 billion euros in private capital.
In almost identical statements, the two banks said after holding extraordinary board meetings on Friday that they had asked their chief executives "to verify the position" of bailout fund Atlante, which is financed by Italian lenders and insurers.
European Union rules which aim to shield taxpayers from the full cost of saving failing banks require private investors - shareholders, senior bondholders and large depositors - to bear losses before any public money can be used.
Italy is keen to avoid such a "bail-in" scenario, fearing this would hurt confidence in the wider banking sector, and wants instead to inject money using an exception to the rules that spares senior bondholders and depositors any losses.
The government is likely to put pressure on healthier banks to provide the private capital demanded by the EU, several banking sources said.
Atlante, hastily set up at the government's behest, already rescued the two Veneto-based banks last year after they failed to raise money on the market. It has pumped in 3.4 billion euros and holds a stake of more than 97 percent in both, but they have since suffered further losses.
Sources close to the fund said it had no intention of stumping up more capital.
"The conditions are not there," a source at one of the fund's main shareholders said, adding that there would be more discussions. Atlante has 1.7 billion euros left, but that has been earmarked to help Monte dei Paschi di Siena and other banks offload bad loans.
Highlighting how hard finding fresh money would be, the head of Italy's biggest retail lender Intesa Sanpaolo - one of Atlante's main contributors - said this week that strong banks should not be forced to spend more rescuing weaker rivals.
"We are in the hands of the government," Popolare di Vicenza Chairman Gianni Mion told reporters on Friday. ($1 = 0.8945 euros) (additional reporting by Paola Arosio, writing by Silvia Aloisi; editing by Alexander Smith)