He is thought to favour becoming Ireland’s next European Commissioner, although the job of President of the Commission is also on offer, and that post will have to be filled at the leaders’ summit in June.
However, Taoiseach Bertie Ahern has not yet given any indication that he will nominate the Munster MEP as Ireland’s next Commissioner and is waiting to see if any other countries mention Mr Cox as a possible contender for the role of Commission President.
Who will be the next Commission President, Mr Ahern replied: “There are a number of very prominent and outstanding people’s names being mentioned and some are being seriously discussed.”
Over the weekend, Mr Cox insisted he was not on the campaign trail and was attending the European Union enlargement ceremonies in Dublin, “not to get a job, but to do a job”.
Mr Cox, who is coming to the end of his two-and-a-half year tenure as Parliament President, could end up as a backbencher in Strasbourg.
Mr Cox has worked hard and closely with the new member states, both in recruiting political parties to the ranks of his Liberal grouping in the Parliament before he became president, and, since then, in helping to integrate them into the institution.
At least three politicians have already started campaigning for the job of Commission President. They are External Affairs Commissioner Chris Patten, Belgian Prime Minister Guy Verhofstadt and former Finnish Prime Minister Paavo Lipponen.
Without the support of one of the bigger political parties in the Parliament, and mainly the European People’s Party (EPP) that has a majority in the institution, Mr Cox lacks a powerful promoter to push his name for the job of Commission President.
Last year the British floated his name but the Irish Government felt this particular endorsement would backfire on the basis that if the British were for him, the French would object.
Mr Ahern has previously said that with the EPP insisting the next Commission President must come from their ranks, there was little point in putting Mr Cox’s name forward.
The jostling for the high-profile job has started in earnest after Mr Patten putting his hat into the ring. The British Tory grandee and former Governor of Hong Kong is well known worldwide and respected by all the EU governments.
He could prove a powerful ally to British Prime Minister Tony Blair when he has to put the EU constitution to a referendum in Britain and help split the Conservative opposition.
The author of the Northern police report that led to the replacement of the RUC with the PSNI, Mr Patten is also liked by the French for his critical stance on the Iraq war.
Mr Cox will announce his intentions in Strasbourg on Wednesday when a delegation from his Munster constituency visits him as he presides over the Parliament’s plenary session for the last time.
Should he run, he could take the second of the three seats in the Constituency after Fianna Fáil’s Brian Crowley. However, he could also find himself in a battle for the third seat with Fine Gael’s Simon Coveney and outgoing FF MEP Gerard Collins.