LATEST: Dr James Reilly expects there to be a new taoiseach by summer
Update 7pm: Michael Noonan says it is up to the next Taoiseach to decide whether there's a role for him in Cabinet.
But the Finance Minister says Enda Kenny might be around for a "little while yet" after what he says was a successful trip to the White House.
Minister Noonan says the outcome of his meeting with Donald Trump shows Kenny has valuable experience in international diplomacy.
But he says whenever a new leader is chosen, it will be up to them to decide if he should stay in cabinet.
Update - 3.40pm: Minister for Housing Simon Coveney has said that he does not believe there has been a change in Taoiseach's position on his departure.
He also said he trusted Mr Kenny to have a process in place for his stepping down and that a motion of no confidence in him as Fine Gael leader is unlikely as it would not be supported.
The party's Deputy leader, James Reilly, told RTÉ's Today with Sean O'Rourke that he expects the country to have a new taoiseach by summer.
Senator Reilly said he "has no doubt" the Taoiseach will deal conclusively and effectively with the leadership issue in the coming fortnight and he does not expect a statement from him at Wednesday's parliamentary party meeting.
Update 12.22pm: A senior Fine Gael Minister says the option of Enda Kenny remaining as Taoiseach while a new party leader goes through a transition period should be considered.
Richard Bruton has again not ruled himself out of running for the job and said that he has absolute confidence in Enda Kenny to step down at the right time.
Earlier, MEP Brian Hayes suggested the new leader be in place by early May and that they could have a period of transition for discussions with Fianna Fáil and Independents while Enda Kenny continued as Taoiseach.
Minister Bruton says it is one of a number of options that should be considered.
“I think the party will have to consider options like that as to whether they bring advantage,” he said.
“They wouldn’t be conventional, but I don’t close my mind to any possibilities.
“The important thing, I think, is that we achieve continuity without a disruption to the progress we’re making, but that … a new Government is formed that has a very clear sense of direction, and I think that’s the thing we need to get right.”
A senior Fine Gael MEP has said that he expects the Taoiseach will stay in office until May at least.
Sean Kelly said this morning Enda Kenny is scheduled to be a guest speaker at a meeting of the EPP bureau here in May, and sometime after that would be "appropriate and acceptable for him to step down".
The issue was revived at the weekend when Mr Kenny told reporters that he cannot leave until a new Government has been appointed in Northern Ireland and the terms of reference for Brexit talks are agreed.