Greece's four big banks - National, Piraeus, Eurobank and Alpha - need to fill a 14.4 billion euro ($15.46 billion) capital hole, a shortfall more than 3.6 times their current market worth.

It will be their third recapitalisation since 2013 when Greece's bank rescue fund HFSF and private investors pumped 28.7 billion euros into the four banks, followed by a second 8.3 billion euro boost by mostly foreign investors last year.

Banks and regulators want to wrap up the exercise by the end of this year, before European Union bank resolution rules go into full effect in January, potentially affecting large depositors in the burden sharing.

But as banks start their book-building to lure private investors to their share offerings, disagreement between Athens and its creditors over tighter protection against primary home foreclosures for problem borrowers is raising concerns.

Greece's left-wing government is keen to avoid indebted Greeks losing their homes - especially at a time when the country is set to provide food and shelter for tens of thousands of migrants and refugees.

But on Monday euro zone finance ministers called on Athens to finalise pending reforms, flagging that without compliance a 10 billion euros sum earmarked for the recapitalization may not be unlocked.

"Our assessment is that the recap will go well and will be concluded successfully," Louka Katseli, head of the Greek banks association told Reuters.

But other bankers admit that the clash between the leftist government and its lenders over home foreclosures is not going unnoticed among investment circles.

"The newsflow on foreclosures created noise and is on the mind of some investors, but overall there is interest in the offerings," said another banker at one of the four lenders. "If there is a fast compromise, it will help."

BOOKS OPEN

Banks will seek to fill as much as possible of their capital needs from private investors. At the very least, they will aim to cover the baseline scenario of a 4.4 billion euro shortfall with private capital before they resort to state aid from the HFSF rescue fund.

The HFSF, or Hellenic Financial Stability Fund, will provide funds when required by buying contingent convertible bonds (CoCos) and new shares the lenders will issue at a ratio of 75 percent CoCos, 25 percent new stock.

Under Greece's third 86 billion euro bailout, 25 billion has been set as a buffer for the HFSF to backstop the recapitalisation.

Piraeus Bank, Greece's second-largest bank which needs to plug a 4.9 billion euro hole, 15 times its market worth, was the first to begin its book-building last week.

"The book will likely close by Friday, at the latest," a banker at Piraeus told Reuters, declining to be named. "There is a good number of binding offers for the new shares and optimism we will meet our goal to at least cover 1.6 billion euros."

Piraeus generated 600 million euros of core equity capital via a debt exchange offer to bondholders last week, leaving it with a need to raise another 1.6 billion euros from the market to fill a baseline capital gap of 2.21 billion euros.

Third-largest lender Eurobank is set to begin its book-building later this week, "likely on Wednesday", a source at the bank said.

Eurobank, 35.4 percent owned by the country's bank rescue fund, had the smallest shortfall - 2.12 billion euros - among the four lenders under the ECB stress test's "adverse" scenario.

Fourth in size Alpha Bank, 66 percent owned by the HFSF rescue fund, is set to open its book on a planned share offering later this week, a senior executive at the bank told Reuters.

"Overall, indications of interest are encouraging," the executive said, declining to be named.

Another banker at National said the bank would soon follow. The group plans to sell its Turkish unit Finansbank to generate capital to fill a 4.6 billion euro capital gap.

"Contacts with investors are going well, management has made stops abroad, in London, New York and other financial centres where there is potential interest," the banker said. "We will not go for a hard underwriting in our offering."

(Editing by Jeremy Gaunt)

By George Georgiopoulos and Lefteris Papadimas