Ministers of the 15 current and 10 future EU member states arrived in Brussels on Monday for a last negotiating session.
Danish Foreign Minister Per Stig Moeller told reporters he was confident of success. "I do think that we will get a treaty Saturday evening." Polish Minister for Europe Danuta Huebner echoed his view.
But back in Warsaw, Prime Minister Leszek Miller said the increasingly tough tone of countries demanding reform of the EU's voting system could create a political crisis.
"We can feel something of a chill. The question is whether this is just acting or toughening their positions. Should the latter be the case, we might be heading for a confrontation and the Brussels meeting could end in a fiasco," Miller told public radio from his hospital bed.
He is currently recovering from injuries received in a helicopter crash last week.
Differences of opinion
The new charter is designed to ensure the bloc can function smoothly after it enlarges from 15 to 25 states next May, expanding eastwards with a total population of 450 million and about one-quarter of world gross national product.
EU foreign ministers have narrowed many differences. But a core dispute over voting powers in EU decision-making remains as intractable as ever.
The draft constitution calls for a simpler voting system taking more account of population size, which Poland and Spain strongly oppose.
Other outstanding issues include the size and shape of the executive European Commission, with small countries pressing for one commissioner per member state.
"We can feel something of a chill ... we might be heading for a confrontation and the Brussels meeting could end in a fiasco"
Polish prime minister
A mutual defence clause which four non-aligned EU member states are resisting is also proving controversial.
There are also serious questions over the powers of the European Parliament and the Commission on the EU budget and economic governance.
Germany and Italy, which holds the rotating EU presidency, said on Sunday they would not give way on the vexed voting rights issue.
German Chancellor Gerhard Schroeder said he and Italian Prime Minister Silvio Berlusconi both agreed they were not prepared to accept the demands of Spain and Poland.
These two are the main beneficiaries of a voting system agreed under the Nice Treaty in 2000, which gives them nearly the same voting power as Germany, despite having populations of about 40 million each, roughly half the size of Germany.
Both Spanish Prime Minister Jose Maria Aznar and Miller have made it clear they are not prepared to give up either.
However, Berlusconi said it would be better to put off a deal rather than reaching an unsatisfactory agreement.
EU leaders are due to wrap up the negotiations by Saturday evening, although the programme allows for possible extra sessions on Sunday.
A 105-member Convention of lawmakers and national representatives headed by former French President Valery Giscard d'Estaing drew up the draft constitution over 16 months in 2002 and 2003.