It comes amid speculation he may be demoted if Leo Varadkar wins
Health Minister Simon Harris says he does not believe the winner of the Fine Gael leadership will be "vengeful" when it comes to Cabinet positions.
It comes after speculation that he would be demoted if Leo Varadkar were to become Taoiseach.
Mr Harris declared himself as a backer of Simon Coveney for party leader early on in the campaign.
Mr Varadkar has seen a lead in support from declared members of the Fine Gael Parliamentary Party.
Voting continues among members of Fine Gael to pick either Mr Varadkar or Mr Coveney as leader.
The first of the 26 polling stations opened on Monday.
Votes 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.
The result is set to be announced in Dublin on Friday June 2nd.
Health Minister Simon Harris says he has been getting on with the job during the leadership race.
"God you guys ask me this all the time - no, I believe whoever wins the leadership election I don't believe wither of these guys to be vengeful.
"I believe them to both be superb candidates.
"I'm backing Simon Coveney because I believe he's the right person for the right time, and I've been very upfront about that.
"But whilst doing that I've also been getting on with my job, and a number of issue in health that needed to be rectified.
"The issue of Orkambi for patients with cystic fibrosis, finally pressing go on the National Children's Hospital, sorting out the National Maternity Hospital".