The Regional Independent group of TDs are set to miss out on a senior Cabinet post as talks on government formation continue.
The group, which could see up to seven of its TDs supporting the next government, is in line to get a super junior role.
This is a Minister of State who attends Cabinet meetings.
Therefore, the breakdown of Cabinet positions is likely to see Fianna Fáil holding eight senior roles to Fine Gael's seven.
TD Michael Lowry said that good progress has been made in the discussions with Fianna Fáil and Fine Gael.
However, he stated that it was not clear yet when these negotiations would conclude, but he anticipated that a new government would present itself to the Dáil on 22 January.
There have also been further talks between the Healy-Raes and the leaders of Fianna Fáil and Fine Gael this evening.
Fianna Fáil negotiator James Browne said it is likely that the text of a programme for government would be published on Wednesday.
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There is expected to be a one-day Fianna Fáil Ard Fheis on Sunday where the party would hear the views of its members, while Fine Gael is planning a series of regional meetings beginning possibly as early as Thursday.
There is speculation that Fianna Fáil TD Jim O'Callaghan could be appointed as the next Attorney General.
The Dublin Bay South TD is a barrister and there is precedent for TDs holding this role, including Declan Costello who was AG from 1973 to 1977.
However, some within Fianna Fáil believe that the role of AG would place Jim O'Callaghan above the political fray and it would impede his ability to vote with the government regularly.