Fianna Fáil TD Marc MacSharry is calling on Taoiseach Micheál Martin to resign as party leader this summer.
It follows the party's poor performance in the Dublin Bay South by-election, where it polled less than 5% - finishing fifth place in terms of first preference votes.
Another party TD, Barry Cowen, is now calling for a special meeting to address the 'alarming' showing.
Deputy Marc MacSharry says Mr Martin should stand down during the Dáil summer recess.
He said: “The sooner the better as far as I’m concerned, quite frankly.
“It’s not my preference he would lead us into the next general election.
"I think a party with a performance of below 5%, with the history, culture and heritage of Fianna Fáil, must address that as a matter of the utmost urgency.”
Deputy MacSharry believes there’d be “no better time” than the upcoming summer recess to deal with the issue.
The Sligo–Leitrim TD noted: “Obviously, it’s a personal view - it would be a matter for the party as a whole.
"But I think we’re now at that stage.”
He suggested Mr Martin has led the party for a “very substantial period” of ten years, and his legacy in terms of the smoking ban and other policies is secure.
The Taoiseach insisted on Friday it is still his intention to lead his party into the next general election.