Mrs Maire Geoghegan-Quinn is believed to be favoured by the Taoiseach, Mr Ahern, as Ireland's new EU commissioner. But agreement has still not been reached with the Progressive Democrats.
Negotiations between the parties were continuing last night, and a spokesman for the Tanaiste, Ms Harney, said the party leaders were likely to "sign off" on the issue shortly.
An announcement about the identity of the commissioner to replace Mr Padraig Flynn will be made within the next three days.
The Taoiseach submitted three names to Brussels last week. They are: Mrs Maire Geoghegan-Quinn, a former minister and currently columnist for this newspaper; Mr Brian Cowen, Minister for Health; and Mr David Byrne, the Attorney General.
Mrs Geoghegan-Quinn caused serious offence to the Progressive Democrats some weeks ago when she described the party as "a threat to democracy" in an article in The Irish Times.
As a result, Ms Harney is understood to favour the appointment of either Mr Cowen or Mr Byrne.
She is also believed to be seeking assurances that Mr Des O'Malley, a former leader of the Progressive Democrats, will be nominated to the European Court of Auditors by the Government.
Mr Ahern yesterday told RTE's This Week he had submitted three names for consideration to the president-designate of the EU Commission, Mr Romano Prodi. Earlier, he had indicated the kind of portfolio the Government would favour. They were agriculture, regional development or social affairs.
Mr Prodi is believed to have favoured the appointment of Mrs Geoghegan-Quinn in order to provide an adequate gender balance within the new commission, due to take up office in the autumn.
Mr Ahern told RTE he had a definite person in mind for the position. But he said he would have to consult with the Progressive Democrats and with members of the Cabinet.
He had already had a number of conversations with Mr Prodi about the issue and he hoped a nomination could be made this week.
Asked if it would be a senior politician, the Taoiseach replied it would be a person of ability and capability, and all three of those nominated would be well able to do the job.
Cabinet sources last night insisted that a by-election held no appeal for the Taoiseach or Government.
There was no certainty Fianna Fail would win a by-election for the minority Coalition Government, even in the Laois-Offaly constituency of Mr Cowen.
As a result, Mrs Geoghegan-Quinn was the firm favourite to secure the position, in spite of resistance from the Progressive Democrats.
It was suggested that Mr Byrne was likely to be appointed to the European Court of Justice if, as expected, Mr John Murray comes back to Ireland as president of the High Court.
The position at the Court of Auditors being spoken of in connection with Mr O'Malley falls vacant next February when Mr Barry Desmond retires.