Greece's prime minister sounded a conciliatory tone ahead of his first summit with European leaders Thursday, hours after the country's talks with eurozone creditors on overhauling its bailout loans broke down.

Though Wednesday's emergency meeting of the eurozone's 19 finance ministers failed to yield a plan of action or even a statement, markets were buoyant on hopes a deal will be reached in time for Greece to avoid a potential exit from the euro.

"We will need to find a solution that respects the positions of all parties, so this agreement will have to be based on the core values of Europe, democracy and the vote of the people, but also on the necessity to respect the European rules," Greek leader Alexis Tsipras said.

Europe has been embroiled in another Greek crisis following the election of Tsipras' radical-left Syriza party last month. The new government was elected on a mandate to drastically reduce the burden of the country's bailout and the associated budget austerity measures, which Tsipras blames in large part for the country's economic woes.

Despite a recent modest return to growth, the Greek economy is around 25 percent smaller than it was before the crisis and poverty and unemployment have swelled. Greece is lumbered by huge debts, which stand at around 175 percent of GDP, and it has repayments this year that it will have trouble meeting without outside help.

Tsipras wants Greecee's current bailout deal, which runs out at the end of the month, to be scrapped and replaced by a new one. But a group of eurozone countries, led by Germany, insists that discussions can only proceed if the current bailout program is extended.

Without the bailout's financial support, Greece faces bankruptcy — and a possible exit from the eurozone, a development that would damage Greece's economy, at least in the short-term, and throw global financial markets into turmoil.

Tsipras said his government was willing to launch further structural reforms "against tax evasion, fighting against corruption, fighting against a state based on clientelism." Those ambitions are likely to be welcomed by the more skeptical members of the eurozone. The question is whether it is enough to ensure a deal.

The European creditors, particularly Germany, are hesitant to give in to Greece too easily for fear of setting a precedent for countries that run up excessive debts. The 240 billion euros (currently $272 billion) in rescue loans Greece is getting come from taxpayers in other countries.

"We have all possible understanding for Greece, but they need to have all understanding for us too," said Dutch Prime Minister Mark Rutte.

Despite the apparent impasse Wednesday, Belgian Prime Minister Charles Michel said "Europe's magic is to always find solutions even when everyone believes it's impossible."

Many analysts think Europe will once again achieve a deal at the last-minute, with Greece agreeing to a bailout extension provided the required budget austerity measures are eased and Greece implements reforms.

"It comes down to: if you change the terms and conditions of a contract, have you created a new contract or do you have the same contract with minor amendments?" said Gary Jenkins, chief credit strategist at LNG Capital. Though Jenkins thinks a deal is "doable," he cautioned that this could develop into a "case where pride really could come before a fall."

Most stock markets were higher Thursday, with Athens' main index up more than 5 percent.

Greece's finance minister, Yanis Varoufakis, laid out hope that progress was possible Monday.

"We understand each other much, much better now than we did this morning, so I think this is a major achievement because, you know, from understanding, the agreement follows," he said. He stressed that Greece would insist on agreeing on a new bailout program, rather than extending the current one, which "has been catastrophic."

Popular support for a change of course is strong in Greece, with thousands taking to the streets across the country Wednesday.

Since winning the Greek general election last month, the Greek government has outlined its broad ambitions but it has still to present concrete proposals.

Besides a new bailout deal, Varoufakis has suggested Greece be granted a "bridging loan" to tide it over for a few months. In return, Greece would commit to further reforms, particularly on how to deal with the country's notoriously inefficient tax system.