Sinn Féin has been called on not to enter government with either of the two big parties and to instead unite in forming a grand coalition on the left.
But the appeal from People Before Profit to Mary Lou McDonald fell on deaf ears, with the Sinn Féin leader arguing that her party must take the step of holding Fine Gael or Fianna Fáil to account by sharing power with either of them.
Earlier, TD Richard Boyd Barrett told Pat Kenny on Newstalk: “I think we have a historic opportunity to move beyond them for the first time in Irish political history.
"They have had their 100 years running the State... in recent years they've left us with a mess in the most basic things: housing, health, the cost of living, the climate and many other things.”
I would appeal to all of the left - including some of the radical left - to build a united front that offers people a genuine alternative to Fianna Fáil and Fine Gael, who I think most people can't tell the difference between the two of them.
But Ms McDonald, said: “Fianna Fáil and Fine Gael, I understand the reticence around them.
"But the actual choice we have to make is whether we are going to allow the two to continue as the dominating and single force in Irish politics and government or whether we are prepared to step up. I am prepared to step up.”
Her party has 19% of support according to a recent poll.
Nonetheless, both Fine Gael and Fianna Fáil have ruled out working with Sinn Féin, given the party's republican past and the role of its Ard Chomhairle, a body - including non-elected officials - which decides policy.
People Before Profit TD Bríd Smith warned that Sinn Féin could betray its voters.
“Every time you talk to a Sinn Féin supporter, the first thing they emphasise is they really want to get rid of a situation whether the country is being run by either Fine Gael or Fianna Fáil.”