The former Taoiseach and Fine Gael leader, Mr John Bruton, has been asked by the party to run in Dublin in the European election next year.
The request was made recently by the leader of Fine Gael, Mr Enda Kenny, although Mr Bruton has not yet decided whether to run in Dublin, Leinster or at all.
Mr Bruton's high profile would give Fine Gael a major chance of winning one of Dublin's four seats in the European Parliament, a key aim of the party leadership.
News of the approach to Mr Bruton emerged yesterday on the same day as the former Fine Gael leader, Mr Michael Noonan, told the party that he had decided not to run in the new South constituency.
Mr Noonan said he had considered running since the party approached him in September. "I didn't have any fear that Fine Gael wouldn't get a seat," he said. "I thought I wouldn't like the lifestyle. I don't think I'd want to be spending several nights a week in Brussels or Strasbourg or wherever the European show settles."
Mr Noonan believed it was possible that the party would run two candidates in the constituency, which comprises the counties of Munster except Co Clare. He said it was possible that the party would run a candidate from Cork and from another county.
The Cork North-Central TD, Mr Bernard Allen, said last night that he had been asked to run. The Cork South-Central TD, Mr Simon Coveney, is also in the frame.
Mr Allen said: "There is a certain level of pressure on me to run. I have to discuss it at home and with my people. I'm slow to make a career change but I'm still giving it thought."
Mr Coveney said: "I wouldn't have run against Michael if he had chosen to run. If I'm asked to run by the party I'd be honoured, but I'd have to give it consideration first." He said a decision to run would have a "huge impact" on his personal life and the direction of his political career.
Senator Paul Bradford, from Mallow, also said he was considering whether to seek a nomination.
"I'm sure I'm one of many. It's not a question of being asked - you'd want to commit to go. It's not a part-time job." Separately, the party's former TD in Dublin South-East, Ms Frances Fitzgerald, has declined the opportunity to be added to the party's Dublin Corporation local election ticket.
There are suggestions within the party that she may now seek a nomination to run in the Dublin constituency in Europe.
Ms Fitzgerald finished third of three candidates who contested two places on the party's Rathmines ward ticket last month.
She was beaten by Mr Brian Gillen and Ms Edie Wynne. Mr Gillen received 56 votes on the first count while Ms Wynne received 45 votes following the distribution of Mr Gillen's transfers. Ms Fitzgerald attracted just 41 votes.