European leaders have been meeting in Brussels for the second round of emergency talks in ten days to hammer out a deal on the top jobs in their union.
Arriving at a summit that could go late into the night, France's President Emmanuel Macron said he was optimistic there would be a breakthrough.
But Germany's Chancellor Angela Merkel was less hopeful, warning the debate would be far from easy to resolve.
The 28 leaders are aiming to agree candidates for president of the European Commission, president of their own Council and a foreign policy chief.
The European Parliament will then vote for its own speaker this week, and a new director for the European Central Bank will be chosen later.
This is likely to be British Prime Minister Theresa May's last EU meeting as she will be replaced as leader on 22 July following her inability to get MPs to agree on a Brexit deal. A spokesman said she would play a "constructive role" and would not abstain from any vote.
Officials are preparing breakfast talks Monday if the summit goes late.
"The way things are presented, they will not be very simple consultations, to put it mildly," Merkel, the bloc's most influential leader, said as she arrived for the talks.
Macron, however, told reporters he expected a "constructive accord".
"I'm not pushing this candidate or that one," he said. "You've never heard me say that I'm stuck on this or that candidate."
He nevertheless reiterated the names of Dutch socialist Frans Timmermans, Danish liberal Margrethe Vestager and French conservative Michel Barnier.
Macron admitted he had been "hostile" to some candidates, a discreet reference to German conservative MEP Manfred Weber.
He stressed that the four candidates nominated by the leaders must include two women and someone from eastern Europe.
Party sources said Weber -- once the candidate of Merkel to head the Commission -- was ready to step aside, though his sidelining will further hurt the chancellor's fragile political standing at home.
This move however would set the stage for the leaders to line up behind Timmermans, vice president of the outgoing Commission, for the top job.
Weber, a veteran MEP with no executive experience, could then count on broad support to be elected head of the parliament.
The German lawmaker may announce this decision later Sunday, party officials said, breaking the logjam.
Merkel discussed this option at Saturday's G20 summit with Macron, Spanish Premier Pedro Sanchez and the Netherlands' Mark Rutte.
One MEP told AFP that summit host Donald Tusk would present Timmermans' candidacy for the Commission job, backed by four major countries.
But, as vice-president of the Commission for the past five years, Timmermans has made enemies in the east of the Union.
Hungary and Poland could stand against the man who spearheaded EU legal efforts to enforce Brussel's vision of the rule of law in their countries.
Hungarian Prime Minister Viktor Orban, in a letter to the centre-right EPP party's leader Joseph Daul, said accepting Timmermans would be a "serious or even historical mistake".
Polish Premier Mateusz Morawiecki said as he arrived: "Frans Timmermans isn't a candidate of compromise.
"Frans Timmermans is a candidate who deeply divides Europe and he certainly doesn't understand Central Europe, he doesn't understand the Europe which is emerging from the post-communist collapse"
If Budapest and Warsaw are joined by more populist or eurosceptic governments, such as Italy's, they could threaten to create a blocking minority.
A successful nominee for Commission president needs the votes of 21 of the 28 leaders, representing 65 percent of the EU's population.
If there is a breakthrough on Sunday, leaders will be relieved to have headed off a damaging split between Paris and Berlin.
Merkel has hitherto backed Weber as the "Spitzenkandidat" or lead candidate of the centre-right European Peoples Party.
The EPP came out on top in last month European parliamentary elections, though with a historically low share of the vote.
Other lead candidates are Timmermans and EU competition commissioner Vestager, who was long seen as a possible compromise pick.
Macron insists the "lead candidate" process for assigning the posts is unworkable and that the 28 national leaders must alone decide.
He has cited Barnier, the EU's respected Brexit negotiator, as a possible Commission head.