Two senior Fine Gael ministers have said they will not run for the leadership of the party while Minister for Finance Michael Noonan has announced he will step down as a Cabinet minister when a new taoiseach assumes office in June.

Tánaiste Frances Fitzgerald said she will not be a candidate in the Fine Gael leadership race but did not indicate her support for either of the front runners, Simon Coveney and Leo Varadkar. Minister for Education Richard Bruton also said he would not be running and gave his backing to Mr Varadkar.

Mr Bruton is popular among the Fine Gael grassroots and his endorsement is an early fillip for Mr Varadkar.

"I do believe that Ireland is changing and that citizens expectations have changed as a result of that and I think that is one of the features that Leo brings to the leadership," Mr Bruton said. "Leo has a clear understanding of that sort of change. I think he is best placed to respond to the challenges we now face and that has been a factor in my decision."

Mr Varadkar is expected to have the support of six Cabinet Ministers including Minister for Public Expenditure Paschal Donohoe and Minister for Jobs Mary Mitchell O'Connor. Minister for Arts Heather Humphreys, Government chief whip Regina Doherty and Minister of State at the Department of Defence Paul Kehoe will also support him.

He also has the public backing of Ministers of State Helen McEntee, Sean Kyne and Joe McHugh as well as TDs Peter Burke, Josepha Madigan, Brendan Griffin, Michael D'Arcy, Alan Farrell, John Paul Phelan, Noel Rock and Colm Brophy.

An Irish Times tracker of the FG leadership race has Mr Coveney trailing Mr Varadkar at present.

Mr Coveney handed in his nomination papers for the leadership contest at Fine Gael headquarters in Dublin on Thursday and called for a positive campaign.

“The party is going to have a very competitive process here that hopefully brings out the best in people, that unites the party, that strengthens the party,” he said. “When this is all over I hope we will come together focused on providing strong Government with big ideas.”

Mr Coveney, who was joined by supporters including Minister for Health Simon Harris and Ministers of State Damien English and David Stanton, declined to say how much he would spend on the campaign.

Mr Varadkar will launch his campaign on Saturday morning and the winner will be announced on June 2nd.

Meanwhile, Fine Gael's deputy leader James Reilly resigned from the position and declared his support for Mr Coveney. “I believe my position is not of consequence, but the future of the party is,” he told RTE’s Today with Sean O’Rourke show.

“I am backing Simon Coveney, I believe he has the temperament for the job. He is hard working, collegiate, not afraid to take on the hard issues...I believe Leo has many fine talents, but for the job ahead Simon is the best leader.”

On what is the first full day of the contest to become party leader, and as a consequence Taoiseach, Mr Noonan said he would not contest the next general election, signalling the end of his political career. Mr Noonan was first elected as a TD for Limerick in 1981.

‘I believe now is a good time for a new Finance Minister to take up office’, said Michael Noonan. Photograph: Eric Luke

“Following the Taoiseach’s announcement yesterday, I want to state that I do not wish to be considered for Cabinet by his successor, whoever that may be,” he said.

“I have been honoured to serve as Minister for Finance since March 2011 and have introduced six annual budgets and one supplementary budget.

“I believe now is a good time for a new finance minister to take up office. Ireland is the fastest growing economy in Europe, the budget is almost balanced and we are on the cusp of full employment.

“I will continue to carry out my duties as Minister for Finance until a successor is appointed...It has been a privilege to represent the people of Limerick since I was first elected in 1981 and I look forward to continuing to do so until an election is called.”

Tánaiste and Minister for Justice and Equality Frances Fitzgerald. Photograph: Cyril Byrne

Earlier, Ms Fitzgerald said she had seriously considered a run but decided against it.

“I have been touched by the encouragement and support I have received,” she said. “But I have decided that entering the contest is not the right decision for me. I look forward to the campaign and the bright future ahead for Fine Gael.”

Ms Fitzgerald said Mr Kenny's "work on behalf of the country and our party has been immense and extraordinary. He strove for the best in a way that was tireless, relentless, enthusiastic and compassionate. His zest for what he does has no equal."

Speaking at an event in Dublin, Taoiseach Enda Kenny said he had no regrets about his decision. He made his announcement at a meeting of the Fine Gael parliamentary party in Dublin on Wednesday after weeks of speculation.

Mr Kenny said he would continue to conduct the work of Government while the contest was under way.