President François Hollande of France said he wanted an agreement, “provided one is possible.” He signaled a strong desire to protect the agricultural subsidies, cherished by French farmers, that make up around 40 percent of the Union’s nearly $1.34 trillion spending plan.
Angela Merkel, the German chancellor, said on arrival at E.U. headquarters that the positions of member states were “still far apart” and suggested that a deal might be out of reach.
“It would be desirable to have a joint result, but we have to wait and work hard, and that’s what I will do,” she said.
David Cameron, the British prime minister, said the amounts of money under discussion were excessive and warned that “if they don’t come down, there won’t be a deal.”
An attempt to reach an agreement on the budget failed in November, creating the need for the leaders to try again now.
Herman Van Rompuy, the president of the European Council, which organizes summit meetings, had intended to start the session in midafternoon and had sought to push leaders to make their complaints clear in a roundtable session rather than being allowed to break into small groups.
But his plan was derailed, and the dinner and first roundtable session were delayed until the evening, as leaders formed the kind of presummit huddles that he seemingly wanted to avoid.
Mr. Hollande, who this week roundly criticized Mr. Cameron for his “à la carte” approach to Europe, after a speech last month demanding a redrawing of Britain’s relationship with the union, was an hour late to a meeting during the afternoon with the British prime minister, Mrs. Merkel and Mr. Van Rompuy.
That lateness was a sign of French displeasure with British demands, including tighter limits on farm spending, according to an E.U. diplomat, who spoke on condition of anonymity because of the sensitivity of the talks.
“The French are playing tough, and they are arguably being more rigid than in November,” said another E.U. official with direct knowledge of the talks who spoke on condition of anonymity because the meetings were going on behind closed doors.
The budget, negotiated every seven years, represents only about 1 percent of the European Union’s total economic output and around 2 percent of total public spending. But it still involves furious horse-trading as leaders focus on getting the best deal for their own countries’ citizens, rather than emphasizing pan-European considerations.
One of the complications in the current round of negotiations has been a call for budgetary rigor from leaders like Mr. Cameron, who says the Union should tighten its belt at a time when many European governments have been compelled to impose stringent budget cuts.
Britain is the strongest supporter of cuts to the part of the budget used to employ 55,000 people, including 6,000 translators, most of them in Brussels, in running the Union’s day-to-day affairs.
But because such uses represent only about 6 percent of the budget, leaders probably will need to prune money that had been foreseen for transport, energy, and digital services, among other projects, that are overseen by the commission, the bloc’s policy-making and administrative branch.
That approach has been met with suspicion, and even hostility, in other parts of the Union.
After the failure to reach a budget agreement in November, another impasse this time would be a severe embarrassment for the Union’s leaders, who have already spent years bickering over how to save the euro. Another failure would also force the Union to use provisional annual budgets that cost more and could delay to 2015 any further chance of a long-term agreement.
To avoid that, leaders were expected to meet into the early hours of Friday morning to hash out a deal that could limit the cash that governments give the Union to €900 billion or slightly higher, but leave the door open to projects requiring additional sums of about €60 billion.
But it remained to be seen if that formula, which would probably satisfy countries that pay more into the budget than they receive, like Britain, would also accommodate the demands of countries like France to maintain generous payments for agriculture and infrastructure.
Even if leaders strike a deal, there may be further battles ahead with the European Parliament, which has the power to veto the overall budget. Some of the most influential figures in Parliament have already signaled that they would reject a budget that foresees spending less on Europe in the years ahead.
Guy Verhofstadt, the head of the alliance of liberals in the Parliament, called on Wednesday for a full-revision clause to be inserted into the budget, so that it could be increased after three years if economic conditions improved.
Martin Schulz, the president of the Parliament, said on Wednesday that he would not approve a budget that ended up widening the overall gap between the amount of cash paid by governments and the somewhat higher amount that includes additional sums known as commitments.