Micheál Martin has come under increased pressure from his party members, councillors and TDs to halt government formation talks with Fine Gael.
John McGuinness, Fianna Fáil TD for Carlow-Kilkenny, who has often been at odds with his party leader, said on RTÉ's Today programme with Sarah McInerney, that he would like to see his party in government with Sinn Féin.
He said there was a consensus within membership of the party and their elected TDs that they should speak to Sinn Féin.
"We have three medium parties now in the Dáil and huge economic and health issues and I think they should form a government and address the economic and social issues we're facing," Mr McGuinness said.
"I see no reason in the economic circumstances that we shouldn't have a government made up of those three parties, that's been my view from the very beginning, and the reason we have no government is because we talk to Fine Gael and the Greens now with nothing emerging.
"It's clear Sinn Féin are locked out of talks in government because Fianna Fáil and Fine Gael have decided to speak to each other rather than the third party the electorate voted for.
"We also said we wouldn't go into government with Fine Gael, now we're faced with something completely different."
Mr McGuinness said that "rather than threaten another election" which he alleges is being whispered about in "the political bubble of Leinster House", the three parties should develop a programme of government to deal with the issues the country is facing.
"It seems ridiculous we would spend so long talking to two other parties for government, rather than three," he said.
When asked if his party would accept Mary Lou McDonald as Taoiseach, Mr McGuinness replied: "They're willing to accept a Taoiseach whose party lost the election, I don't think that's acceptable.
"Leo Varadkar opened the door to the Green Party having a term as Taoiseach too - politics has changed considerably and we should listen to the electorate and move on.
"I see no difficult in working with other parties if were serious about getting the country moving again.
"Exclusion does not work and we encouraged reversal of that in the north.
"We have to be straight up with the electorate and follow their decision from polls and form a government, and cut the nonsense.
If we continue to grapple with this old style of politics and don't bring about reform, we have failed.
Likewise, Fianna Fáil Cllr Deirdre Kelly sent a statement to the press on Wednesday afternoon, lamenting her party: "is no longer the party of great leaders like De Valera, Lemass or Lynch.
"It is now a generic remake of the centre left social democratic parties that had flourished, but now die, throughout Europe."
Ms Kelly, who represents the party on Cork County Council said there "are very genuine reasons for concern about a Fianna Fáil/Fine Gael coalition", and questions how compatible the two parties would be.
"Fianna Fáil now faces its greatest challenge. It can coalesce with Fine Gael and end up being effectively wiped out at the next election.
"If Fianna Fáil is to survive, the party needs to get back to the ordinary people of Ireland and really listen to them.
"That will not be possible in a coalition agreement with Fine Gael."
A number of grassroots groups of local Fianna Fáil members have written the the party's elected representatives and party headquarters in recent weeks in an effort to discourage moves to form a government with Leo Varadkar's party.
Rumours of a possible leadership against Mr Martin began after the party's poor performance in the February election, however the leader himself said recently he was not phased by the commentary.