The result of the contest will be announced on June 2nd
Housing Minister Simon Coveney says he can still win the Fine Gael leadership contest.
His rival, Social Protection Minister Leo Varadkar, has taken a substantial lead in support from declared members of the Parliamentary Party.
The two men have launched leadership bids to replace Enda Kenny.
Speaking to Pat Kenny, Minister Coveney said: "I wouldn't be in this contest if I didn't think I could win it, I wouldn't be in this contest if I didn't think that I was the best person for the job.
"I'm not used to being an underdog actually in politics, or in life for that matter.
"Normally I'm starting in front and trying to stay there - so this is an unusual place for me to be in".
Mr Coveney said the party leadership is about two things.
"First of all it's about choosing the person who is best qualified for the job - in other words who can handle not only being party leader, but also taking over the Taoiseach's office... and secondly, and actually more importantly to me and to the Fine Gael party: we asking the question what is Fine Gael politics about in 2017? What's the value system that drives this party? What are the priorities that we are going to change the country with".
Votes for the leadership will be weighted in accordance with the Fine Gael electoral college rules.
This will see 73 members of the Parliamentary Party accounting for 65% of the total vote, almost 21,000 party members accounting for 25% and 235 local representatives accounting for the remaining 10% of the vote.
Mr Coveney believes he can still win the contest: "If I'm going to win, I have to change the mind of a relatively small number of those Oireachtas members - about six.
"This is a new system for electing a leader - it's a college system... If we get up above 30 votes and if we can bring Leo Varadkar back to 40 or less, well then we have a real contest here and we believe we can win it."
"I was nominated before Leo was - that was a presumably a strategic thing on his behalf and ours.
"Clearly Leo got a bit of momentum, a number of people came out behind him, and then the big middle ground in the party - who may well have been breaking both ways - decided they saw one person moving ahead and they wanted to get on side".
Asked about recent mudslinging in the campaign, in which Fine Gael TD Kate O'Connell called Mr Varadkar's supporters "choirboys", Mr Coveney said: "I'm not going to get in to making judgements on my colleagues - people will make decisions based on the facts that they see in front of them."
"The vision of this party, in my view, needs to be to represent everybody - whether you're in a sleeping bag on Grafton Street or whether you're somebody who's creating 1,000 jobs, you have to be equally important to Fine Gael as far as I'm concerned in terms of the contribution you can make to society".
He also confirmed he has not abandoned the prospect of people paying for water.
And asked if he would he object to serving in a Cabinet under Leo Varadkar as Taoiseach, Mr Coveney replied: "No, I wouldn't.
"The one thing I would say to everybody is that I believe that the process we're going through in this party will strengthen this party".
Voting for almost 21,000 Fine Gael party members will take place from Monday May 29th to Thursday June 1st. The Parliamentary Party will meet and vote in Dublin on Friday June 2nd.
The result will be announced at the Dublin count centre on Friday June 2nd.