Heads of the new, 25-member European Union have begun a summit in Brussels aimed at agreeing a constitution and a new executive head of the organisation.
Prodi is due to step down in October
A voting rights dispute among other things scuppered a summit six months earlier, but some heads of government went into the talks with optimism.
Just as difficult will be the choice of successor for Romano Prodi, the Italian President of the European Commission.
Talks halted on Thursday evening, but are due to resume before midnight.
Earlier, both French President Jacques Chirac and Irish Prime Minister Bertie Ahern, who is chairing the talks, told journalists they felt they were close to an agreement.
The heads of government are having to confront an intensely personal problem as most of the potential candidates are around the table themselves, the BBC's Tim Franks notes.
Face to face
The British have been outspoken in their opposition to the leading contender, Belgian Prime Minister Guy Verhofstadt.
Guy Verhofstadt, Belgian PM
Jean-Claude Juncker, Luxembourg PM
Antonio Vitorino, EU Commissioner
Pat Cox, European Parliament President
Bertie Ahern, Irish PM
Chris Patten, EU Commissioner
A surprise candidate has also emerged in the form of a UK commissioner, Chris Patten, whose name has been forwarded by the conservative group in the European Parliament, known as the European People's Party.
However, France is one of several states which oppose Mr Patten's candidacy and prefer Mr Verhofstadt.
Mr Chirac told reporters on Thursday that he did not think it was a good idea to have a candidate from "a country which doesn't take part in all European policies".
The remark appears to be a veiled reference to the UK and Denmark, both of which have yet to join the European single currency, the Euro.
Mr Prodi has said there is no obvious successor to him.
At the December summit in Brussels, Poland and Spain blocked the agreement because they would have lost more favourable deals secured previously.
The political situation in both countries changed recently - Spain has a new government and Poland is in the grip of a political crisis.
Some leaders arrived predicting success
However, a group of about 10 smaller EU states, including Finland and Austria, say they will reject new proposals for the EU's system of taking decisions.
The new plan calls for measures to be approved when supported by states that make up 55% of the states in the EU and 65% of the total population.
The small states think that formula gives too much influence to countries with large populations.
This dispute risks becoming a repeat of the stand-off which led to the failure of the constitution talks last time, the BBC's William Horsley reports.
Other new proposals by the EU presidency include:
- trimming down the European Commission from the expected 25 members - one for each country - to 18, but only from 2014
- agreeing a minimum of six seats per country in the European Parliament - a move aimed at placating smaller EU nations who fear their voice will be lost in the 732-seat assembly.
'No natural successor'
Mr Prodi is stepping down as president of the European Commission when his five-year term ends in October.
Apart from Mr Verhofstadt and Mr Patten, names touted for the job include Mr Ahern, Austrian Chancellor Wolfgang Schuessel, former Nato chief Javier Solana, Portuguese commissioner Antonio Vitorino and European Parliament President Pat Cox.
But Mr Prodi told a news conference on Wednesday that there was no majority yet for any candidate.
He said the nominee needed a qualified majority from the council of EU leaders.
"So far we don't seem to have achieved that," he said.