Mr Donohoe insisted yesterday that he would not contest the position, but confirmed he has been approached by colleagues to put his name into the mix for the leadership.
The news comes as Fine Gael descended into chaos yesterday with widely diverging views emerging as to what Mr Kenny should do.
Supporters such as junior minister Dara Murphy say that given the uncertainty brought about by Brexit, Mr Kenny should remain in office for up to 18 months. There was a growing sense Mr Kenny’s desire for a 12-week reprieve could be granted to allow him “space to step down with dignity”.
However, one backbench TD, Dublin-Fingal TD Alan Farrell became the first person to call outright for Mr Kenny’s departure, saying his position has become “untenable”.
“Unfortunately, I no longer have confidence in the ability of Enda Kenny to lead Fine Gael and I believe his position is now untenable,” he said in a statement.
His intervention has followed similar calls from TDs Noel Rock and Pat Deering, the latter has threatened to put down a motion of no confidence in Mr Kenny next week.
Mr Farrell’s intervention came in for strong criticism from one of the main contenders for the leadership, Housing Minister Simon Coveney.
Speaking to the Irish Examiner Mr Coveney said: “I believe what Alan said is wrong. This is absolutely the wrong time for this type of intervention. The Taoiseach is his own person and I don’t think he will respond to ultimatums. People who are demanding timelines by next Wednesday, I think, that represents a very small minority in the party.”
The chair of the Fine Gael parliamentary party was last night forced to write to members to warn them to stop calling for Mr Kenny’s departure as it is damaging to the party.
Martin Heydon emailed TDs and senators stating he is “greatly concerned that the Taoiseach and leader of our party is not being shown the respect his service and his office deserve”.
Mr Heydon continued: “I ask - in the best interests of a united Fine Gael and in the interests of common decency - that members of the parliamentary party refrain from making further public utterances about our party leader. Such comments ultimately hurt our party.
“Any comments should be kept for next week’s parliamentary party meeting which will take place as usual at 5.30pm next Wednesday,” he told members.
Mr Donohoe restated comments he made to the Irish Examiner: “I am very lucky to be where I am, I want to continue doing the job I love. I will not be a candidate.”
But, the Irish Examiner has learned that there is a growing desire for Mr Donohoe to reconsider his stance and remain open to contesting the race. This view is held by Mr Kenny and Mr Noonan and others who feel Mr Donohoe would be best suited to steer the party into the future.
Mr Donohoe himself admitted he has been approached by colleagues about the leadership position: “Yes, conversations like that have happened in recent months within the party. Colleagues and individuals have talked to me because while we have respected the work Enda has done, we are aware of the need to look to the future. I want to play a leadership role in Government. I don’t want to put my name forward should a vacancy arise.”
Minister for Foreign Affairs Charlie Flanagan has also ruled himself out of the for the leadership of Fine Gael.
Responding to reports that a generational wipe-out would occur at Cabinet once the leadership changes, Finance Minister Michael Noonan said such talk was premature.