European leaders are urging Greece's conservative New Democracy party to form a new government promptly after its victory in the weekend elections.
European Union politicians said Greece now had to enact key reforms and a German government spokesman said the time is not right to grant Greece any leeway or additional time on its reform commitments.
The centre-right New Democracy party - led by Antonis Samaras - claimed almost 30% of the vote and nearly 130 seats in his country's second general election in six weeks.
Mr Samaras said Greeks had voted to stay in the euro, and called for a "national salvation government", the BBC reports.
He wants to form a government as soon as possible.
The leader of the anti-bailout Syriza party, Alexis Tsipras, which came a close second, agreed Mr Samaras should be first to try to form a coalition.
Germany said it viewed the result as a decision to "forge ahead" with reform.
With 80% of votes counted, interior ministry projections put New Democracy on 29.9% of the vote (128 seats), Syriza on 26% (71) and the socialist Pasok on 12.4% (33).
Coalition talks were expected to start later on Monday, with the most likely ally being the Pasok party.
Mr Samaras says: "The Greek people voted ... to stay on the European course and remain in the eurozone.
No more adventures
"There will be no more adventures. Greece's place in Europe will not be put in doubt."
"The sacrifices of the Greek people will bring the country back to prosperity," he promises.
He also says Greece will "honour its obligations".
Mr Samaras wants to press ahead with spending cuts demanded by the country's international creditors, the BBC reports.
European leaders have warned that if a new Greek government rejected the bailout, the country could be forced to abandon the single currency.
While the radical-left Syriza and other smaller parties have opposed the bailout, New Democracy and Pasok said they would keep it in a renegotiated form.
Germany's Finance Minister, Wolfgang Schaeuble, says he views the election result as a decision by the Greek people "to forge ahead with the implementation of far-reaching economic and fiscal reforms in the country".
Eurozone finance ministers say in a statement that such reform is "Greece's best guarantee to overcome the current economic and social challenges and for a more prosperous future (for) Greece in the euro area".
They expect representatives of the European Commission, the European Central Bank and the International Monetary Fund - the so-called Troika - to return to Athens as soon as there is a Greek government in place.
Germany Foreign Minister Guido Westerwelle suggests Athens might be given more time to comply with its obligations.
"There cannot be substantial changes to the agreements, but I can well imagine talking again about timelines," he said.
Mr Tsipras, the Syriza leader, congratulates Mr Samaras on his apparent victory, and says he has the right to try to form a government.
But he appears to rule out joining such a coalition, saying Syriza would "not sacrifice our position" of opposition to the austerity programme, the BBC reports.
Pasok leader Evangelos Venizelos proposed a broad four-party coalition including New Democracy, Pasok, the Democratic Left and Syriza.
Sunday's election was was called after a vote on 6 May vote proved inconclusive and attempts by each of the main parties to form a coalition government failed.
The vote was watched around the world, amid fears that an exit by Greece from the euro could spread contagion to other euro zone members and deepen the turmoil in the global economy.