Greek leftist leader Alexis Tsipras has turned down a mandate to form a coalition government, saying he was preparing for a second poll after a "painful" electoral defeat for his Syriza party.

The conservative New Democracy Party stormed to victory with 40.8 per cent of the vote in Sunday's poll which sent the leftist Syriza into a tailspin, polling 20.1 per cent - the result of many voters' disenchantment with its radical, anti-establishment style.

On Tuesday, Tsipras told President Katerina Sakellaropoulou it was impossible to form a coalition government.

"I have no reason to hide that the electoral result is a painful shock - it was unexpected," Tsipras told reporters outside the presidential mansion, apologising to Syriza supporters.

"I take full responsibility for this result, but in my dictionary, that means standing and fighting."

Greek Prime Minister Kyriakos Mitsotakis, leader of New Democracy, declined to seek a coalition on Monday, paving the way for a second vote on June 25 that he hopes his conservative party will win outright.

Without New Democracy, opposition parties do not have enough seats to form a ruling alliance, and all party leaders have indicated they will not hold exploratory talks.

Syriza refers to the second vote as a "final battle" which has yet to come.

In a televised statement on Monday, Tsipras said Syriza's primary responsibility was to "prevent the prospects of an almighty and uncontrollable ruler-prime minister" and ensure the presence of the left in Greece's political landscape.

Syriza's defeat revealed a split left.

Two small leftist parties, set up by former Syriza members, did not make it into parliament.

During the pre-election period, Syriza tried to persuade the Socialist PASOK party, which finished third in Sunday's election, and leftist parties, including the Communist KKE, to back it in a coalition government.

But after its defeat, Syriza accused them of turning their back on its efforts to form a broader alliance against the conservatives.

PASOK will also get a mandate to form a coalition government before the president appoints a caretaker government that will lead Greece to a second vote.

That election will take place under a system of semi-proportional representation, with a sliding scale seat bonus, increasing the chances of an outright win for Mitsotakis's party.

All parties are eligible to run again.