BRUSSELS — Prime Minister Alexis Tsipras of Greece asked on Wednesday for a summit meeting of eurozone leaders that would allow him to make his case for easier terms on sorely needed aid to help his country avoid bankruptcy.

Without new rescue money by July, Greece could default on its debts and throw the 19-member eurozone into another period of chaos. There could also be a domestic upheaval in Greece similar to last summer, when the country had a referendum on the terms accompanying its third bailout, followed by snap general elections.

Mr. Tsipras’s latest plea is similar to one he used a year ago, when he won plaudits at home for negotiating one-on-one with Chancellor Angela Merkel of Germany.

However, at a summit meeting in July, the left-wing Mr. Tsipras ended up accepting a third bailout in five years on even worse terms than if he had settled the matter earlier. And European leaders are in no mood to engage in yet another round of brinkmanship with Greece while facing an array of other challenges, including the prospect of Britain’s leaving the European Union and pressure from an assertive Russia.