It will bring to an end a six-year spell as Taoiseach and a campaign will now begin to a new Fine Gael leader and Taoiseach before June 2.
Varadkar has said he has more support among those yet to declare while Coveney, whose father was a Fine Gael minister during the 1980s and brother is chief executive of foodmaker Greencore, has said the race is "very, very close".
Speaking on Newstalk Breakfast earlier, the Public Expenditure Minister described Varadkar as "the most substantial and capable politician of my generation".
Ministers Charlie Flanagan and Paul Kehoe, as well as TD Fergus O'Dowd, have also backed Varadkar.
Judge declares Prince's 6 siblings the heirs to his estate
Prince died April 21, 2016, of an accidental overdose of fentanyl, a synthetic opioid drug 50 times more powerful than heroin. No one will ever know what Prince wanted done with his estate after he passed away because he didn't leave a will.
His supporters hope to narrow Mr Varadkar's lead, or pull ahead of him, before the series of hustings that will take place next week and mark the engagement between the candidates and rank-and-file members.
"Having inherited a country and an economy that was decimated by mismanagement, Enda's strong leadership and ability to make hard decisions was recognised both at home and overseas and saw our global reputation restored and strengthened", he said.
Two candidates have so far declared an interest in becoming the next party leader and, by default, the next prime minister for the remainder of what is widely considered to be a fragile, minority government. "That is precisely what we need right now and I believe Simon Coveney will deliver on this", Mr Harris said.
Around 21,000 Fine Gael members will be entitled to vote with TDs and Senators accounting for 65 per cent of the vote, delegates 25 per cent and local councillors the remaining ten per cent share.
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Observers looks for signs in Chinese state media that Beijing might finally cut Pyongyang off financially and politically. US President Donald Trump has recently made remarks to this end which have caused confusion in South Korea.
"I will continue to carry out my duties as party leader in an acting capacity, until my successor is elected through the Fine Gael leadership election process", he said.
Mr Coveney set out his stall for the leadership during his first interview of the campaign on RTÉ's Prime Time last night.
Enda Kenny speaks at a luncheon hosted by the Chamber of Commerce of Metropolitan Montreal during his visit to Montreal, Quebec, Canada May 4, 2017.
It was a short but historic meeting of the Fine Gael parliamentary party, when an emotional Enda Kenny announced he would be standing down as leader, writes political reporter Elaine Loughlin.
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In other news, insider SANDA PAREZANOVIC purchased 35 shares of the stock in a transaction that occurred on Tuesday, April 18th. According to analysts Coca-Cola Company (NYSE:KO)'s minimum EPS for the current quarter is at $0.56 and can go high up to $0.6.
Mr Kenny delayed his resignation on a number of occasions this year.