Former taoiseach Bertie Ahern does not believe a large number of Fianna Fáil TDs would back a coalition with Sinn Féin.
He said the party is still coming to terms with its performance in the election, and that Sinn Féin is in 'the driving seat' when it comes to forming a government.
With all Dail seats in the 33rd Dail now filled, attention has shifted to what form the next government will take.
When the new Dáil meets for the first time, Fianna Fáil will have 38 seats, Sinn Féin 37 and Fine Gael 35.
Sinn Féin leader Mary Lou McDonald yesterday said she is beginning talks with other left-leaning parties such as the Greens and the Social Democrats, with the goal of putting together "something new and constructive".
Former Fianna Fáil leader Bertie Ahern told Newstalk Breakfast that there's unlikely to be support for a Sinn Féin alliance if Ms McDonald fails to put together a left-leaning government.
Explaining what he believes will happen next, he said: "In the first instance - the remainder of this week - I think it will be Mary Lou, as she has the right to do, [going out] to see if she can get her coalition of the left... talk to Labour, Social Democrats, People Before Profit, the Greens and the independents.
"I can't see that being a success... but it could be. There's a sizeable number of rural independents - at least ten - of a group that are like-minded enough... who got strong votes."
He suggested Sinn Féin is in the 'driving seat' going into the government formation talks, adding that he believes a Fianna Fáil and Fine Gael arrangement would lead to a 'large reaction from the people'.
If no left government is formed, that would leave a Fianna Fáil / Sinn Féin coalition - with the likely support of a smaller party - as one of the most obvious paths to a majority.
However, Mr Ahern argued: "I do not think that there will be a sizeable number of Fianna Fáil people saying 'well as soon as Mary Lou is finished her project, if she can't get it, we're ready to jump in'... I don't think that will be there.
"I think the challenge for Fianna Fáil is how they reorganise themselves and build themselves for the future - because there always is the issue of the next election."
He also suggested he would find it very hard to see how a 'revolving Taoiseach' arrangement would work if that prospect comes up during negotiations.