While Mr Varadkar is expected to personally launch his bid on Saturday, Mr Coveney fired the opening salvo on the steps of his party headquarters and declared he was not going to engage in sound bites and glib put-downs.
Aoife Heffron, spokesperson for BoyleSports, said: "With Enda Kenny's retirement coming into effect, we have seen an influx of bets on Leo Varadkar, who was once priced at 11/10 to take over as Fine Gael leader, but now heads the betting at the short odds of 2/5".
FOLLWING confirmation that Taoiseach Enda Kenny has stepped down as Fine Gael leader effective from midnight last night, BoyleSports have reported that Leo Varadkar is the firm 2/5 favourite to succeed him as party leader.
"I want to lead a party and a country that can heal numerous divisions that have opened up over the last decade or so".
Frances Fitzgerald, the tánaiste, said that she was "touched by the encouragement" to enter the contest but that it was not the right decision for her to do so.
Minister and Donegal TD Joe McHugh has announced he is backing Social Protection Minister Leo Varadkar to be the next Taoiseach.
Car plows into crowd of pedestrians in Times Square
With the advent of summers, it is peak tourist time in one of the most popular destinations in the world. Terrorists have used vehicles to crash into crowds in London, England and Nice, France in the past year.
Minister Coveney has held his current position since May 2016; he served as the Minister for Agriculture, Food and the Marine between 2011 and 2014. Junior ministers Damien English and David Stanton are also in the camp.
The Finance Minister said he did not want to be considered for another cabinet job and would not stand again for the Dail.
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. "Ireland is the fastest-growing economy in Europe, the budget is nearly balanced and we are on the cusp of full employment", he said.
Here, Clare's Fine Gael representatives are considering their options on who will lead their party into the future.
He has always made himself available to members of the Parliamentary Party and to me at any point, and I have brought up a number of groups from both Longford and Westmeath to meet him in his office to discuss various policy issues.