Tánaiste Frances Fitzgerald has said she is seriously considering putting her name forward as a candidate in an expected Fine Gael leadership contest in the coming months.
She said last time there were four candidates and it is healthy for the party to have a good debate leading up to the nomination for the new leader, but for now the Taoiseach is in place.
Earlier, Minister for Health Simon Harris said he will not be a candidate in any upcoming leadership race.
Speaking on RTÉ's Today with Sean O'Rourke, Mr Harris said the time is not right for him to run.
The minister said he was flattered to be asked by a number of people but the timing was not right.
He said he had a challenging ministry to deal with and he did not have the experience for the role.
Mr Harris said he was ambitious for his party and country and he was ruling himself out to end any distraction.
He declined to say who he would back for Fine Gael leader but said once the Taoiseach announced he was stepping down, he would announce who he would support for leader and that he would campaign for them.
Enda Kenny last month told a meeting of the Fine Gael parliamentary party that he will deal with his future effectively and conclusively after St Patrick's Day.
He is due to make an official trip to Washington DC for the St Patrick's Day celebrations.
Asked about comments by former minister for justice Alan Shatter suggesting Mr Kenny has a casual relationship with the truth, Mr Harris said that that is not the Taoiseach he knows.
The Fine Gael executive council met last week and agreed a number of matters in relation to a leadership contest.
Under the party rules, the leadership is decided by an electoral college which gives members of the parliamentary party 65% of the votes, ordinary members 25% and councillors 10%.
It is understood the executive council agreed that there will be 28 venues around the country where councillors and party members can vote.
This will occur over a number of days, the details of which has yet to be finalised.
Members of the parliamentary party will vote separately.
It was agreed that there will be no postal voting in the leadership contest.
The executive council will meet again when a contest is announced.