Former minister for agriculture Barry Cowen has requested a special meeting of the Fianna Fáil parliamentary party to discuss the result of yesterday's 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.
In a letter to party colleagues Mr Cowen described the party's worst ever Dáil election result as "both a shock and alarming but strangely not hugely surprising."
He said the matter cannot be brushed aside or ignored and called for a summer meeting to allow people attend in person.
The Laois/Offaly TD said by-elections offer opportunities for the electorate to send messages and it is good manners to respond to these messages.
"Our parliamentary party, its leadership and executive personnel of the party should meet during the summer to reflect, discuss and take on board these messages," he said.
Fianna Fáil TD for Clare @CathalCroweTD tells @SaturdayRTE he does not believe Micheál Martin will be leading the party into the next general election.— Saturday RTÉ (@SaturdayRTE) July 10, 2021
'There will be a new leadership by then' @rtenews pic.twitter.com/GL5pirej0E
Meanwhile Fianna Fáil TD Cathal Crowe has said he would expect the party to have a new leader going into the next general election.
Speaking on RTÉ's Saturday with Katie Hannon, the Clare TD said it was "just a personal view", but added that "a lot would share that view".
However, Mr Crowe said that Taoiseach Mícheál Martin is "doing a pretty good job in a time of crisis".