Mayo Fine Gael TD Michael Ring has said Enda Kenny's successor will have to show the same commitment and stamina that the Taoiseach did.
The prime minister leads a minority government under a deal with the opposition Fianna Fail party, which has veto power over Fine Gael's decision over who to nominate as Kenny's successor.
Kenny, prime minister since 2011 and leader of his party for nearly 15 years, is expected to be replaced by either Leo Varadkar, the social protection minister, or Simon Coveney, the housing minister.
In a statement, Enda Kenny revealed he will "retire as leader from midnight tonight 17th May 2017".
Kenny - who had already announced he would not lead Fine Gael into the election - said he would remain prime minister during the contest, due to be concluded on June 2, and during subsequent talks with lawmakers backing the government.
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He said: "I especially want to thank my wife Fionnuala, our children, my siblings and their families for their understanding of my work, and indeed for accepting the many intrusions of politics into family life in the interest of building our country".
But when asked whether Enda Kenny would be addressing the leadership issue at this meeting, the Fine Gael Press Office and Government Press Office both refused to comment on the matter.
"I could not have engaged as I did without that base".
The front-runners to replace Mr Kenny as Fine Gael leader and subsequently as taoiseach are Leo Varadkar, Social Protection Minister, and Simon Coveney, Housing Minister. "Did his country much service", Labour Party leader Brendan Howlin, who served with Kenny in coalition until past year, said on Twitter.