EU leaders did not reach agreement on Friday morning (21 June) on who should lead the European Commission and other key institutions and will meet again at the end of next week.

"There was no majority on any candidate," Tusk said, adding that EU leaders agreed that there needs to be a package reflecting the diversity of the EU.

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Tusk will continue consultations, including with the European parliament.

Leaders will gather again next Sunday (30 June) evening.

"We want to agree before the new [EU] parliament meets," German chancellor Angela Merkel told reporters after the meeting.

Merkel emphasised that an institutional conflict between the council of member states and the parliament must be avoided.

"At no point do we want to see a crisis with the parliament, it would not be good for the EU as a whole for the years to come," she said.

The European Parliament has been arguing leaders to pick a person to run the new EU commission from the lead candidates put forward by the political parties.

However, there is no parliamentary majority backing any of the hopefuls.

The centre-right EPP, which emerged as the largest party in the elections last month, has pushed the German Manfred Weber, and the Socialists, who came in second are fighting for Dutch commissioner Frans Timmermans.

Danish commissioner Margrethe Vestager ran as a liberal candidate.

The centre-right, the Socialists, the liberals and the greens have been working on a coalition agreement in the parliament but have been unable to agree on a candidate for the commission top job.

The parties want to come to an accord by next Tuesday, but it is unlikely they will be able to come up with a common candidate by then.

Under the new leadership of Spanish MEP Iratxe Garcia, the Socialists have dug in to fight for Timmermans, while the new liberal leader Dacian Ciolos has also not endorsed Weber.

EPP officials see this move as a hostile one against Weber, coordinated by the Socialist government in Madrid and French president Emmanuel Macron in Paris, whose MEPs supported Ciolos' bid as the liberal group leader.

EU leaders discussed the new EU leadership during their dinner on Thursday night under secrecy, with phone signals being scrapped and aides waiting outside the meeting room.

"I noted with pleasure, amusement and happiness that I am not that easy to replace," quipped the EU commission's current president, Jean-Claude Juncker, at the end of the meeting.