EU Commission President Jean-Claude Juncker said the issue was still far too complicated to reconcile with so many competing bids and candidates lined up to replace him, Council President Donald Tusk and four other top jobs.
He held out little hope it would be easier next week for the new summit.
"I don't expect that but it has to be done," Juncker said.
German Chancellor Angela Merkel and French President Emmanuel Macron had said it was better to wait a few more days and continue talks than to make hasty decisions.
The bloc's leaders gathered after May's EU parliament elections, determined to quickly wrap up the politically charged process of choosing new EU leadership.
Under EU rules, the member countries choose who will run the commission, replacing Juncker. The European Parliament must endorse that choice. But the assembly has been trying to play hard ball, insisting that only the party leader candidates who ran in last month's elections should be eligible for the post.
But that principle has been questioned by Macron, whose party joined a new free-market liberal group bolstered by the European Parliament election results, and set him on a collision course with Merkel, whose Christian Democrat group suffered losses.
The EU is responsible for coordinating common policies on sectors ranging from the single market to agriculture, from competition issues to immigration. In addition to the presidency posts, a new EU foreign policy chief must be chosen, as well as a chairman of the European Central Bank. The new parliament will also choose its own president when it sits on July 2.
For weeks now, each nation has sought to jockey their favorite candidate into the best position to exert the most clout for the next five to eight years.
The job responsibilities are huge: Tusk and Juncker negotiate with the likes of U.S. President Donald Trump or Chinese leader Xi Jinping, while the head of the ECB can set monetary policy for the 19 nations who use the shared euro currency.
The leaders of EU institutions are supposed to impartially represent the interests of all member nations on the global stage and at home. But patriotism set in as officials from individual EU countries pushed candidates from their homelands to rule the roost of the bloc's population of 500 million and the world's biggest economic alliance.
Beyond national concerns, there was geography too. There needs to be a mix of big member states, like Merkel's Germany, and small ones, like Juncker's Luxembourg. The west and east, north and south also need to be represented in the EU somehow. The outgoing group of EU officials was lopsidedly Italian, with Tajani holding the parliament top post, Mario Draghi head of the ECB and Federica Mogherini the EU foreign policy chief.
Then there is the political competition between the conservative European People's Party, the Socialists, free-market liberals and even the Greens to make sure enough of their concerns shine through. Many feel that even though the EPP group is the biggest in the EU parliament, having Tusk, Tajani and Juncker all in such high positions was too much, especially since that group lost seats in the EU's May election.
This time around, many feel that women need to be better represented at the top of the EU's hierarchy.
Top candidates include current prime ministers Mark Rutte of the Netherlands, Stefan Lofven of Sweden and Andrej Plenkovic of Croatia. Others mentioned for top EU jobs include Brexit negotiator Michel Barnier of France, Greens leader Ska Keller of Germany, Lithuanian President Dalia Grybauskaite, EU first vice president Frans Timmermans of the Netherlands and Margrethe Vestager, the EU's competition chief since 2014.
Lorne Cook contributed
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