Richard Bruton and Frances Fitzgerald have ruled themselves out of the race
Housing Minister Simon Coveney has officially launched his campaign for the Fine Gael leadership.
Minister Coveney was the first minister to formally declare his candidacy this morning.
The other expected candidate, Social Protection Minister Leo Varadkar, has yet to speak to the press.
A 'Campaign for Leo' website has gone live this afternoon, with a note that the campaign will officially launch on Saturday.
Speaking outside Fine Gael headquarters earlier today, Minister Coveney said: "For me, I'm in politics to be positive, to be constructive".
He pledged "honesty" and "big ideas", saying the party now needs to "rebuild a strong society in Ireland".
He told reporters: "I want to lead a party and lead a country that can heal many of the divisions that have opened up over the last decade or so.
"Unfortunately, politics in recent times has been about cynicism, has been about negativity, has been about dividing communities across Ireland [...] I want to be a Taoiseach that can change that."
Oh there's a more serious version too pic.twitter.com/drkcVp5oYo— Páraic Gallagher (@paraicgallagher) May 18, 2017
Education Minister Richard Bruton ruled himself out of the race outside his department shortly after midday, and confirmed that he is endorsing Leo Varadkar.
Minister Bruton explained: "I believe he is decisive, he is reforming. He has great capacity to deliver and has shown a track record of doing that."
Meanwhile, Finance Minister Michael Noonan has announced that he does not want to be considered for the next Cabinet.
He will continue in his ministerial role until a successor is appointed, and will no run for re-election as a TD.
Earlier, the Tánaiste and Justice Minister ruled herself out of the leadership race.
Frances Fitzgerald says she did seriously consider it, and was "touched by the encouragement and support" she received.
However, she has decided that entering the race is not the right decision for her.
A leadership contest was initiated after Enda Kenny stepped down as party leader yesterday.
Mr Kenny will continue his duties as Taoiseach in an acting capacity, until a successor is elected.
Votes will be weighted using a complex system that gives the most decisive say to the parliamentary party ever.
While the party's 73 TDs, senators and MEPs enjoy the strongest voting strength (65%), councillors and rank and file members of the party will also have a say.