Mr Kenny stood down from the helm of Fine Gael at an emotionally-charged meeting of party colleagues in Leinster House in Dublin after delaying the announcement for several months.
Under Fine Gael party rules, TDs and senators account for 65% of the votes, while councillors account for 10% and general party members make up 25%.
A new leader will be announced on Friday, June 2.
"I will retire as leader of Fine Gael effective from midnight tonight", Kenny said, opening the way for a swift leadership contest.
He was humbled by the early support he secured but insisted the contest was not over.
Meanwhile, Leo Varadkar described him as an "extraordinary Taoiseach and Fine Gael's most successful leader ever".
Mr Coveney, from Cork, has also held a number of ministries but faces weekly criticism of his efforts to tackle the country's unprecedented housing and homelessness crisis.
"He took the tough decisions and because of those the decisions the country is back to where we are today", she commented.
"We've been through a really hard recession which is now over but the wounds caused by the recession are still real and many of them are open".
"We can repeat the mistakes of the past, allow ourselves to lose control of public spending, fuel the property market. or this time we can get it right". He has led the Fine Gael for 15 years and is its first leader to win re-election to office.
The race is set to be dominated the two declared candidates: Varadkar, 38, and Coveney, 44.
Each session will also include a question and answers session and will be live-streamed online.
Helen McEntee, another junior minister in Mr Varadkar's camp, said: "Look at the fresh new approach he has taken to politics, look at how he interacts with people, look at his vision for a more forward-thinking and open Ireland".
Mr Coveney said that it "would be doing a disservice to the party" to consider halting his campaign and that "people need to understand and hear in detail what the candidates who are intending to lead the party and country are about".
Leadership hopeful Varadkar, now Minister for Social Protection, visited Donegal last week. Following this he was the Minister for Defence between 2014 and 2016.