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20 May 2017, 12:15 | Max Haynes
"I do not wish to be considered for Cabinet by his successor, whoever that may be", Noonan said in a statement.
The governing centre-right party's new leader will be in place by the end of 2 June, Kenny said.
Her predecessor David Cameron said Mr Kenny "was a strong leader for Ireland, a great partner for the United Kingdom, and remains a good friend".
Minister for Social Protection, Leo Varadkar and Minister for Housing, Simon Coveney are the clear frontrunners in the race.
Mr Kenny said he would give his successor a "brief but appropriate" time to hold talks with parties and independents propping up the Republic's minority government.
The father of the Dail leaves a legacy of steering the Republic out of a crippling recession, albeit with an agenda of austerity.
The decision comes following a disappointing General Election result a year ago, which saw Kenny's party fail to secure a majority in the Irish Parliament and criticisms about the way in which he has handled a police scandal. The focus will now turn to a number of Ministers who have yet to outline whom they will support.
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While Noonan and Kenny have been widely praised for their management of the economy, their Fine Gael party lost a quarter of its seats at elections past year as voters complained they were not feeling the recovery.
Leadership hopeful Varadkar, now Minister for Social Protection, visited Donegal last week.
Clare's Fianna Fáil TD says he hopes whoever wins the leadership of Fine Gael will honour the confidence and supply agreement - Timmy Dooley says both candidates should be well-positioned to tackle the pressing issues.
Senator Tim Lombard is on Mr Coveney's top table for the campaign, while Cork North Central TD Dara Murphy and Senator Jerry Buttimer have also expressed their support publically.
"I also believe Minister Varadkar will help stabilise our country, deal with the challenges of Brexit and help Donegal and the border counties in the months and years ahead".
A party leadership contest has officially begun.
Fine Gael lawmakers and councillors make up 75 percent of the vote, with ordinary members accounting for the balance.
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