By Jan Strupczewski

BRUSSELS (Reuters) - Euro zone countries are prepared to keep financing Greece until the country forms a new government, whether one emerges from Sunday's election or if new elections have to be held next month, euro zone officials said on Thursday.

Greek Socialist leader Evangelos Venizelos began efforts to form a government on Thursday, but Greece already looks to be moving inexorably towards new elections after most voters rejected an EU/IMF bailout, plunging the country into crisis.

"I expect an announcement of new elections in Greece by Sunday at the latest," one euro zone official said.

Venizelos is the third and last political leader to attempt to form a government after Sunday's vote, in which the majority of Greeks voted for fringe parties opposed to the bailout.

"My understanding is that a second election in Greece could be by mid-June. We have the means to support Greece through the end of June," a second euro zone official said.

Asked if the funding could be disbursed even though there would be no proper Greek government to enforce the conditions of the bailout, the official said: "No one would expect a non-existent government to be passing legislation."

In remarks to his parliamentary group, Venizelos acknowledged he only had a slim chance of forming a government, after efforts by the leader of the conservative New Democracy party and the head of the radical-left party also failed.

The radical-left party led by Alexis Tsipras, 37, came second to New Democracy in the vote, forcing Venizelos's PASOK into third place, on a promise to throw out the 130-billion-euro bailout agreed with the EU and IMF in March.

"We will provide enough funds for Greece to stay afloat for as long as the political decision is clarified," the first euro zone official said.

"There is no use letting them default in the middle of things. That is what yesterday was all about - giving them enough money to stay afloat and not induce new chaos if people are not paid, but not giving them more than the bare minimum to discourage parties which say that 'we can do whatever we want and they will still save us because it is in the EU's interest.'"

The euro zone's temporary bailout fund, the European Financial Stability Facility (EFSF), agreed on Wednesday to immediately disburse 4.2 billion out of a 5.2 billion sub-tranche of the bailout to Athens, retaining 1 billion to be paid out in June as necessary.

The EU is now Greece's biggest creditor and if Greece defaults, it will be euro zone taxpayers who will take a hit.

"I think that in the collective thinking we have already accepted that we will not get 100 percent back," the first official said, indicating that it may be necessary at some point in the future to write down the value of official loans to Greece.

(Reporting By Jan Strupczewski; editing by Luke Baker)