Theresa May attended the talks in Brussels which continued until the early hours of the morning. A new summit will now be held on 30 June.
The three lead candidates (called Spitzenkandidaten) to succeed Jean Claude-Juncker as the next European Commission president failed to receive enough support from the leaders of the EU’s member states.
French President Emmanuel Macron said: “The three Spitzenkandidaten, the three names were tested by [European Council Chairman] Donald Tusk and he considered that they had found no majority on either of these three names.”
Mr Macron had wanted to block German Chancellor Angela Merkel’s choice for the role – Manfred Weber.
Mr Weber is a deputy head of Mrs Merkel’s centre-right sister party, CSU.
Mr Macron said there was “no majority for Mr Weber”, adding: “I have nothing against a German candidacy.
“I said it and it wasn’t a joke – had the chancellor been a candidate, I would have supported her, because I think she has the qualities, the skills to be a very good president of the Commission.
“It is not what she wants, I respect that very deeply.”
Mr Macron is said to favour Michel Barnier – the EU’s Brexit negotiator – for the post.
Danish EU commissioner Margrethe Vestager and European Commission First Vice President Frans Timmermans are also in the running for the role.
A new Commission chief must get the backing of the majority of national leaders and also the support of the new European Parliament, which sits for the first time on 2 July.
Mrs Merkel said: “We suggest the person, but we definitely don’t want to see a crisis with parliament, to realise that what we suggest is not being accepted.
“This wouldn’t be good for the work of the Commission in the next five years.”
EU leaders are also attempting to appoint a new head of the European Central Bank and president of the European Council – a post currently held by Donald Tusk.
