Among Mr Kenny's final official overseas engagements will be a two-day trade mission to Chicago and the centenary commemoration of the start of the World War One Battle of Messines. Government chief whip Regina Doherty is expected to follow suit later today.
"He did his country much service, and I wish him the very best in the next chapter of his life".
The Coveney camp also made a big play on his own admission that he had asked to be Minister for Housing in the midst of an unprecedented homelessness and housing crisis.
Noonan also said he will not stand for re-election as a member of parliament.
In his statement he the people of Mayo for their support.
Many consider ministers Leo Varadkar and Simon Coveney to be the favourites to fill the vacancy.
David Cameron took to Twitter to pay tribute, posting: "Enda Kenny TD was a strong leader for Ireland, a great partner for the United Kingdom, and remains a good friend".
As the only Fine Gael government TD in this constituency, I believe it is important for the people of this area to have access to the taoiseach and for me to have a good working relationship with the candidate.
The debates will be followed by four days of polling in 26 locations nationwide with the new leader confirmed on 2 June.
Speaking on Newstalk Breakfast earlier, the Public Expenditure Minister described Varadkar as "the most substantial and capable politician of my generation".
Campaigning: Leo Varadkar speaks to the media on Leo Street in Dublin on Friday. It is, as they say, all to play for.
"In health something that has been going for years is the development of the National Children's Hospital and under his leadership of health he finalised the location".
"He understands, as he said last week, the unique circumstances of Donegal, cut off from the rest of the North for most of the last century by partition, and cut off geographically and in other ways from the rest of the State".
"Leo is a person whom I hold in the highest regard, I believe he has the experience, skill and integrity to provide the leadership our country needs and deserves".
Deputy Fergus O'Dowd has said that he is also backing Varadkar for the role.
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