Simon Coveney says the issue of the party leadership will have to be looked at in the "not too distant future"
Leo Varadkar has said he would love to lead Fine Gael "whenever a vacancy arises".
The Minister for Social Protection says it is something he has wanted to do since he joined the party as a teenager.
However, he claims there is no vacancy for the top job at the moment.
The comments came ahead of the vote on the Bill on fatal foetal abnormalities, which has put pressure on Enda Kenny's leadership.
Independent Cabinet ministers are set to go against Fine Gael and vote in favour of the legislation this afternoon, despite the Attorney General advising it is unconstitutional.
Mr Kenny has previously indicated he will not lead his party into the next general election, but intends to serve his full term as Taoiseach.
Speaking to KFM earlier, Minister Vardakar said he would love to lead the party.
"Of course I'd love to lead my party - I've been dedicated to it since I was 17 or 18 years old," he said. "But that's something you have to consider at the appropriate time, when a vacancy arises".
Meanwhile, Simon Coveney says the issue of the leadership of Fine Gael will have to be looked at in the "not too distant future".
However, the Housing, Planning and Local Government Minister says right now is not the time.
He says the discussion by some at the parliamentary party meeting last night was "pragmatic" - but to start speculating now was unhelpful as ministers have a job of work to do.
"I don't see why we need to have a long, drawn out process that actually distracts a Government away from the core issues it needs to deliver on for Irish people," he suggested.