Update 12.20pm: Simon Coveney has said the issue of the leadership of Fine Gael will have to be looked at in the "not too distant future".
But the housing, planning and local government minister has said right now is not the time.
He has said 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: “There will be a time in the not too distant future, I expect, when leadership is going to have to be discussed in Fine Gael.
“But I don’t see why we need to have a long drawn out process, that actually distracts the Government away from the core issues that it needs to deliver on for Irish people.
Earlier Leo Varadkar has said he would love to lead Fine Gael, and has wanted to since he joined the party as a teenager.
However the Minister for Social Protection said there was no vacancy for the top job at the moment - despite the fact that Enda Kenny's leadership has come under pressure over today's abortion vote.
The fatal foetal abnormality bill has caused a rift between Cabinet colleagues and triggered a backlash against Enda Kenny.
Independent Ministers are ready to ignore his advice and vote in favour of the legislation this afternoon.
Speaking to KFM, Minister Varadkar said: "I have thought about it. I'd love to lead my party in the future, but that's something you have to consider at the appropriate time when a vacancy arises.
"I'd have to assess the level of support I might have and see if people would buy into my vision of how the party might change and how the country might change."