Kildare South TD Martin Heydon, chairman of the Fine Gael parliamentary party, has paid tribute to the outgoing Taoiseach Enda Kenny.
Leo Varadkar has said he is "humbled" by the support he has received in the Fine Gael leadership race so far.
Mr Kenny stood down as Fine Gael leader at midnight however he will remain as prime minister until the position is filled.
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If elected party leader, Mr Varadkar said he wants to reach out to those who are not supporting the party and he also wants to make it a campaigning party.
According to the Irish Times leadership tracker, Varadkar holds 31% of the available support from the party TDs and senators, while Coveney commands just 17%.
Kenny led Ireland through a difficult recovery from the 2008 global financial crisis, which hit the debt-fueled "Celtic Tiger" economy particularly hard.
The 66-year-old had been under pressure to resign from factions within his own party that were dissatisfied with his leadership.
"He is the most substantial and capable politician of my generation", Mr Donohoe told Newstalk.
Mr Kenny became known as the "Father of the Dáil" due to being the longest serving Taoiseach in Irish history.
National broadcaster RTE said last week that of those in the parliamentary party who had confidentially declared their support - around half - 23 favored Varadkar, the son of an Indian immigrant who is Ireland's first openly gay minister. If the two parties can not agree, it would trigger national elections and Kenny would stay in office as head of a caretaker government.
"I have asked that the Fine Gael executive council expedite this process and to have it concluded by close of business on Friday 2 June".
"I hope now Leo goes on to win this Fine Gael contest and help lead our country after the enormous progress of this Government over the past six years".
Social Protection Minister Leo Varadkar is seen as one of the frontrunners to take over from Mr Kenny as leader of FG.
"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".