But France's President Emmanuel Macron has clashed personally with the Netherlands' Mark Rutte and Austria's Sebastian Kurz, accusing them of putting the entire European project in danger through their "egotism" and threatening to storm off if they do not listen to his advice and that of Germany's Chancellor Angela Merkel.
Rutte told reporters he had come to the summit to take care of his own country, not make friends with other leaders and "go to each other's birthdays" — Merkel turned 66 on Friday, the first day of the talks — but said that, despite the tension, a deal was "very close".
"We haven't yet found the way through, and it could still fail, but I'm more optimistic than I was last night when, at one moment, I told myself, 'It's over'," Rutte told reporters.
Leaving the talks just after dawn on Monday, Macron said "it's not over, but it's tough" and back in Paris his economy minister Bruno Le Maire declared that "a deal is possible and a deal is necessary."
According to witnesses, at one point Macron thumped the table, berated Kurz for leaving to take a call and accused Rutte of behaving like former British premier David Cameron.
Cameron took a hard line at EU summits but ended up leading his country into a referendum to quit the bloc.
In a bid to break the deadlock, Michel came up with a new proposal on Monday morning, which a diplomat said could be a "route to a deal". "A plan with 390 billion in grants but smaller rebates for the Frugals," a European official explained.