Fianna Fáil's Director of Elections for the Dublin Bay South bye-election Jim O'Callaghan has ruled out a motion of no confidence in party leader Micheál Martin.
However the Dublin Bay South TD said he supports party colleague Barry Cowen's call for a meeting to discuss the bye-election results.
He asked: "The question we need to ask is 'why are we not connecting with the electorate'?"
Speaking on RTÉ's This Week, Mr O'Callaghan said he has not been approached by anyone to sign a motion of no confidence in Taoiseach Martin.
"That hasn't happened and I wouldn't," he said.
Mr O'Callaghan said it is unlikely that Mr Martin would be leading the party into another election if the Government runs to the end of its term in 2025.
He said he is interested in leading the party, but did not want to undermine the current leader.
"I am interested in leading Fianna Fáil. I believe it's an honour and a privilege but I'm not going to do anything just to undermine the leader or give the media an impression that there's something going on when I don't think there is," he said.
Earlier, the Minister for Education has said she "absolutely" thinks Micheál Martin should lead Fianna Fáil into the next general election.
Speaking on RTÉ's The Week in Politics, Norma Foley said there is "job of work to do" after Fianna Fáil's performance in the Dublin Bay South by-election.
The party's candidate, Deirdre Conroy, received just 5% of the share of first preference votes.
Labour's Ivana Bacik, who topped the poll after the first count, was deemed elected on the ninth and final count.
Ms Foley admitted the result was "not a good day for Fianna Fáil."
Mr Cowen has requested a special meeting of the Fianna Fáil parliamentary party to discuss the result of the by-election, while Clare TD Cathal Crowe has said he would expect the party to have a new leader going into the next general election.
Asked if the Taoiseach can lead the party into the next election, Ms Foley said: "Absolutely, Mícheál is a very experienced, solutions-focused leader.
"We see the benefit of that in Government, we also see the benefit of that in Fianna Fáil," she said.
Ms Foley was also asked if she though there are ten TDs in party who would sign a motion against Mr Martin, to which she said "Not at all".
The Minister for Public Expenditure and Reform, Michael McGrath, has said party members need to work together, support the Taoiseach, and "have unity of purpose".
Writing on Twitter about the party's bye-election loss, he said: "The voters have given us a message and we will learn from this. We are listening. There will be an opportunity for everyone in the party to have their say and give their views. That said, this is not a time for us to be talking about ourselves."
He said Fianna Fáil faces a "monumental challenge".
"To succeed, we need to support our Taoiseach, our team, work together, have unity of purpose and show the public that their concerns are our priorities at all times," he added.
Fianna Fáil TD James Lawless said it "wouldn't be a bad time to call it a day" if the Micheál Martin stepped down as party leader after his time as Taoiseach.
Speaking on the Sunday Edition programme on 103.2 Dublin City FM, Mr Lawless said: "There's a transition when Fianna Fáil hand over the Taoiseach's office to Fine Gael and to my mind that would be a sensible time to start to look at the future, or even before then, and plan for that to be a natural break point."
He said the party could "possibly do with" a younger leadership team with a new dynamic at that stage.