The two party leaders met for about 15 minutes this evening
The acting Taoiseach Enda Kenny says he 'regrets' that Fianna Fáil turned down an offer to serve in a partnership government.
"I believe that this decision is a serious mistake and one which was driven by narrow party interests rather than the national interest", Mr Kenny said.
"Ireland needs a stable and lasting government to meet the many national and international challenges facing the country".
"Fine Gael's preferred option of a full partnership is the best option for providing the necessary stability and it is very regrettable that Fianna Fáil has rejected this historic opportunity", he added.
The Fianna Fáil leader has told the acting Taoiseach he could not go into government with Fine Gael.
Micheál Martin has Mr Enda Kenny he may be in a position to support a Fine Gael-led minority government.
The two men met for about 15 minutes after the Fianna Fáil Parliamentary Party decided to reject the offer of a full and equal partnership government.
Speaking this evening, Mr Martin also questioned the motives of Mr Kenny in the coalition offer.
Simon Coveney said it was not in the national interest, that the offer remains on the table and he hopes we are not heading for another election.
And Leo Varadkar says he is disappointed at both Fianna Fáil's decision, and the comments of Mr Martin.
Fianna Fáil rejected the offer from Fine Gael for an equal partnership government.
The decision was made after a three and a half hour meeting of the parliamentary party today.
Within minutes of the offer being made by Mr Kenny last night, it started to emerge that Fianna Fáil TDs were not in favour of it.
And as they arrived this morning at Leinster House for a parliamentary party meeting, there was no appetite for it - with just a few saying 'maybe' it should be explored.
Inside, TDs spoke at length - saying no partnership arrangement was possible when the electorate had rejected Fine Gael and Enda Kenny.
So the talking with Independents will continue for now at least.
But there is a growing sense among those Independents that any minority arrangement would not provide a stable government for five years.
Both parliamentary parties had gathered this morning to consider the offer.
Fine Gael's meeting passed a motion giving the idea their full support.
Catherine Byrne, acting chairperson of the Fine Gael Parliamentary Party, said: "Fine Gael TDs and Senators overwhelmingly supported a partnership government, as the best way to provide a stable and lasting government to deal with the issues concerning people and the challenges facing the country".
While Helen McEntee, secretary of the parliamentary party, added: "This is an historic offer, representing seismic change in the political landscape. I believe now more than ever we need to put the people first".
Last night's face-to-face meeting between the leaders of the two biggest parties came 40 days after the general election result.
The offer was made after the two leaders both failed in their efforts to be elected Taoiseach yesterday.