Fine Gael leader Enda Kenny is seeking a meeting with acting PD leader Mary Harney about forming an alternative government.
He has already been in touch with his Labour counterpart Pat Rabbitte and Green Party leader Trevor Sargent, and has also had "some initial contact" with Independent TDs.
A Fine Gael spokesman said no meetings were scheduled at the moment with other parties or Independents, but Mr Kenny would be holding discussions in Dublin today with senior colleagues and staff members in his own party.
Speaking on RTÉ News, Mr Kenny said: "I expect to have some talks with Pat Rabbitte. I also want to follow through on my intention to seek discussions with the other party leaders, Mary Harney and Trevor Sargent, and each of the Independent members over the next number of days, in the next 10 days or so, because I believe that I really have a mandate for change here."
He continued: "For any party to gain 20 seats from an Opposition platform is actually unprecedented in Irish politics. I think that's been a phenomenal success and from that perspective there's a real mood for change there and I intend to build on that mandate."
Fine Gael director of organisation Phil Hogan TD told The Irish Times: "Fine Gael, Labour and the Greens have 77. Fianna Fáil have 78, so it's practically a draw and it's a question now of seeing what the attitude is of the other Independents and parties to see if an alternative [ can be found]."
Commenting on the latest turn of events at the Mahon tribunal as it affects Taoiseach Bertie Ahern, Mr Hogan said: "It would certainly cause some disquiet, I am sure, in terms of what he said before the election and what is now emerging after the election, and it will be interesting to see how those matters will clarify themselves in the next couple of days."
Mr Hogan added: "Anyone that looks at this very dispassionately would have to look at the options that are available and, if 60 per cent of the people of Ireland have voted against Fianna Fáil, that doesn't give them a mandate of a majority-vote nature to govern."
A Labour Party spokesman said the parliamentary party would meet tomorrow and the national executive was meeting on Thursday, but he described these as "routine" events.
He added: "Bertie Ahern is in pole position as regards forming a government and that's still our expectation."
Labour TD for Dún Laoghaire Eamon Gilmore said: "It's obvious that we are going to explore the possibilities of forming a government. In the aftermath of every election, people talk to each other, sound each other out."
But he added that "there hasn't been any contact" with Fianna Fáil.