Paschal Donohoe has confirmed he will not be putting his name forward for the leadership of Fine Gael.
Mr Donohoe also said that he fully supports Enda Kenny as leader.
The Minister for Public Expenditure and Reform was speaking at a ceremony to mark the 100th anniversary of the battle of the Somme.
Mr Donohoe said that there is no leadership contest at the moment and the only thing he is interested in is implementing the Programme for Government.
Tánaiste and Minister for Justice Frances Fitzgerald and Minister for Social Protection Leo Varadkar joined in Mr Donohoe's sentiments this afternoon when they said that Enda Kenny should be allowed to step down at a time of his own choosing.
Mr Donohoe is the fifth Fine Gael minister to publicly express support for the Taoiseach and Fine Gael leader.
Simon Coveney and Charlie Flanagan have also said they support Mr Kenny.
Government Chief Whip Regina Doherty yesterday called on Mr Kenny to clarify his intentions on how long he will remain as leader.
Speaking on LMFM, Ms Doherty said to do so would allow the party "get back to normal".
Without a definitive timeline, she said Fine Gael TDs and senators will be giving a different opinion on the issue every week.
But speaking on The Claire Byrne Show earlier today, Ms Doherty said that any decision to be made is the Taoiseach's to make in his own time.
Ms Doherty said she was loyal to Mr Kenny and she expressed "utter confidence" in him.
Her comments yesterday that the Taoiseach should give a timeline of his retirement plans were misconstrued, she said.