Dick Roche, Irish minister for European Affairs, said leaders were "tantalizingly close" to a deal on the constitution, six months after their last attempt to agree collapsed in acrimony over voting rules and other issues.
"The atmosphere is very, very positive," he said.
"The issues that are outstanding are those of fine detail," he said, adding all that was left was "final finessing of the endpoints."
British Prime Minister Tony Blair's spokesman told The Associated Press a deal was "nearly there.
"We are still dotting the i's and crossing the t's," he said on customary condition of anonymity.
He said it was unclear whether leaders would now return to the divisive task of choosing a European Commission president, or put that off until a later date.
An EU diplomat, also speaking on condition of anonymity, said the leaders had resolved the voting rights dispute by agreeing that measures must be passed by a majority of countries, plus one, representing 65 percent of the EU's population.
The diplomat said leaders expected to finish the talks shortly.
During the two-day summit, leaders have sought to overcome divisions over a host of issues, notably a new voting system, tax harmonization and even whether to refer to God in the text.
After last weekend's European Parliament elections, marked by record low turnout and a surge in support for Euro-skeptic parties, leaders are eager to reach a deal to show they are capable of making key decisions with the minimum of fuss.
"It is necessary to show people that the European Union can function," Slovenian Prime Minister Anton Rop told AP. "To fail a second time would be a very very bad message."
Once a deal is reached, all 25 nations must formally ratify the treaty and several are planning a referendum, including Britain.
Outgoing European Parliament President Pat Cox, at least, had high hopes. "If we can agree it, we can sell it," he said.
The EU just last month expanded from 15 to 25 members, with many of the newcomers in Eastern Europe and from behind the former Iron Curtain.
The expansion has forced to the surface old tensions within the union about immigration and economic policy. In Britain, the prospect of a European super-state frightens those who maintain a cool attitude toward the mainland.