Brian Hayes has said Easter would be a good time to "kick-start" the Fine Gael leadership campaign.
Taoiseach Enda Kenny caused some controversy when he suggested in New York he would not stand aside until an executive was formed in Northern Ireland and the separate Brexit negotiations begin.
He told reporters these issues “take precedence” over giving a timetable for his departure as Fine Gael leader and Taoiseach.
However, on Monday, Fine Gael MEP Mr Hayes predicted there will be a new leader of the party in May, with a transition plan in place.
Mr Hayes told RTÉ's Today with Seán O'Rourke show also said there is no plan for "a blood letting" within the party. "We are a difficult enough shower to lead at the best of times, so fair play to him. Everyone wants a smooth transition – he will do it in his own time."
His party colleague (and deputy leader) Senator James Reilly said he expected Mr Kenny to no longer be Fine Gael leader is saying “by the summer”.
Mr Reilly said the Taoiseach had made it very clear he will deal with the issue of his departure on his return from the USA. “There is no rush. I don’t believe for one minute that he will be there next year. There’s no way he’s hanging around. He will do the right thing. We don’t need a heave.”
Mr Reilly said that the Taoiseach would not have his legacy undone by a disorderly departure.
The two leading candidates to replace Mr Kenny dismissed concerns the Taoiseach will continuously delay his departure date.
Minister for Housing Simon Coveney and Minister for Social Protection Leo Varadkar said the comments by Mr Kenny on Friday in New York did not indicate any significant change to his previous statements.
Mr Coveney said nothing Mr Kenny had said surprised him, stressing it would be unrealistic for him to walk away from those important issues.
The Minister told The Irish Times: "Some people are suggesting this is an attempt to push things back. I do not think so. Putting together an assembly in the North and making sure Ireland has its stamp on the EU negotiations are two very important issues. The Taoiseach has been involved in those and he wants to make sure they are on the right track.
“People are trying to read into lots of things and find stuff that is not there. They see conspiracy theories around the corner but anyone who knows the Taoiseach is not surprised by what he said.”
Mr Kenny had told his parliamentary party he would make his departure date known when he returned from his St Patrick’s Day trip to the US.
However, the Taoiseach is not expected to address the issue at the parliamentary party meeting this Wednesday, but sources close to him say he will make a statement within weeks.
Mr Varadkar said the Fine Gael leadership contest should not distract from Mr Kenny's successful trip to the United States. He said that should be the focus of the Fine Gael party rather than a debate on their next leader.
Many within the party have now moved towards a summer departure date for Mr Kenny.
One senior figure told The Irish Times the timing of his resignation was irrelevant as long as he made the date known to the party soon.
Separately Fine Gael TD Pat Deering, who had considered tabling a motion of no confidence in the leader, said he believed a contest during the Easter break would be ideal.
Standing aside
He said nobody believed Mr Kenny would land in Dublin airport from the United States and announce he was standing aside.
The Carlow-Kilkenny TD said: “That was never going to be the case. An early departure date in May would have been my preference. It is possible still.
“The boat has sailed. It is not a matter of if any more, but when.”
Mr Deering confirmed he would not be proposing any confidence motions, insisting he had suggested one some time ago with the sole ambition of initiating the discussion on the leadership.
The chairman of the Fine Gael parliamentary party, Martin Heydon, said the timeline had not altered despite Mr Kenny's comments.
Mr Heydon said he believed the Taoiseach would stand aside when the timing was appropriate.
