European Union leaders may meet later this month to try to decide who should become the new president of the European Commission.
Bertie Ahern (left) wants the issue resolved before 30 June
Officials in Ireland, which holds the EU presidency, say 27 June and 30 June are being considered for a meeting.
At the Brussels summit last week, EU leaders reached a constitution deal, but failed to agree on a nominee for the role of commission president.
The current president, Romano Prodi, is due to step down later this year.
A number of names have been discussed and debated, some causing friction at the summit.
Two front runners for the post, the Belgian Prime Minister, Guy Verhofstadt, and the EU commissioner for external relations, Chris Patten, have both pulled out in recent days.
France and Germany had backed Mr Verhofstadt, but the UK opposed him on the grounds that he could be too eager to make the EU more federal.
Mr Verhofstadt later said he was "no longer available" for the post.
French President Jacques Chirac objected to Mr Patten's candidacy saying 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" - such as the single currency.
The Irish Prime Minister Bertie Ahern said he hoped to resolve the issue before his country's presidency ends on 30 June.
Mr Ahern's name has been touted by the Irish media after he told a television programme he was not in the running "at this stage".
Among other possible candidates are Danish Prime Minister Anders Fogh Rasmussen; President of the European Parliament Pat Cox; Portuguese Prime Minister Jose Durao Barroso and French Foreign Minister Michel Barnier.
Luxembourg Prime Minister Jean-Claude Juncker has repeatedly said he would rather stay on in his job than run the EU executive, but he is still tipped as a contender.
EU foreign policy chief Javier Solana has also been suggested in some quarters.
He told reporters: "There are people who could do it better than I, but if there were a consensus request by the political leaders of the EU, it would be very difficult for me to say no."
But on Monday he also made light of the situation saying: "I have just had an operation on my knee so I am running for nothing."
Germany has been hardening its position on the issue.
EU foreign policy chief
Jose Durao Barroso, Portuguese PM
Michel Barnier, French Foreign Minister
Anders Fogh Rasmussen, Danish PM
Antonio Vitorino, EU Commissioner
Pat Cox, European Parliament President
Bertie Ahern, Irish PM
Government spokesman Bela Anda said: "I do not want to formulate preconditions, but it can hardly be imagined that the commission president will come from a country not represented in the core areas of EU bodies and its institutions."
Such criteria would exclude nominations from more than half of the EU, including the UK, Ireland, Denmark, Sweden or any of the 10 new member states which joined on 1 May.
UK Foreign Secretary Jack Straw insisted the nominee could come from states that were not in the euro or other "core areas".
"We follow the constitution of the European Union at the moment and it is very clear that the president of the Commission can be drawn from any member state," he told reporters.
Italian Foreign Minister Franco Frattini, following a meeting with Mr Straw, said EU leaders must unite behind a candidate for the job, or the post would be unworkable.
"Choosing a president without unanimous consensus on the part of all the states would mean putting this president of the commission in a very difficult situation," he said.