Independent TDs are seeking as many as three junior ministerial roles as well as a super junior appointment as part of Government negotiations, but the role of Ceann Comhairle remains up for grabs.
It is understood that the two main parties have indicated a willingness to give those in the regional group of Independents two minister of state roles as well as a super junior role, which would allow that person attend Cabinet.
However, the group of Independents is pushing for more than this with one indicating that “we might get one more” before adding that “it’s hard to keep everyone happy”.
The group of non-party TDs is also seeking the backing of Fianna Fáil and Fine Gael to ensure Verona Murphy is appointed Ceann Comhairle next week.
But one source indicated that Micheál Martin "still has some work to do" to convince party members to back her for the powerful position, with veteran TD John McGuinness entering the race alongside outgoing Ceann Comhairle Seán Ó Fearghaíl.
Mr McGuinness filed his nomination papers to fill the position as the Dáil's chair on Thursday.
He told the Irish Examiner that Mr Martin was aware of his decision.
He said he had been nominated by "a smattering" of people across parties and said reform of the Dáil is needed.
"I'm in the Dáil for 27 years and know the workings inside and out. I know the changes that need to be made to allow members go front and centre to make it more democratic. I see reform as being a centrepiece of this work. I see the members being central to that."
The vote to fill the Ceann's chair will be held on Wednesday when the 34th Dáil sits for the first time.
Fianna Fáil sources said they believe that the party's TDs would "much rather" vote for one of their own than Ms Murphy.
While some said the Wexford TD had "annoyed" Fianna Fáil and Fine Gael in previous dealings across the Dáil chamber, one party TD said that it was "not personal".
"Why wouldn't we want to see one of our TDs elected to the role?" they said.
Others questioned whether Mr Martin and Taoiseach Simon Harris could even promise the role to anyone given that it is one of the rare instances of a secret ballot in Irish politics.
In 2020, the Government's nominee for Leas Ceann Comhairle Fergus O'Dowd was beaten by Galway TD Catherine Connolly in a shock result.
A Fianna Fáil TD questioned whether Mr Martin would leave himself open to a similar result.
Meanwhile, the Social Democrats will continue discussions with Fianna Fáil and Fine Gael around government formation next week, the party has decided.
Deputy leader Cian O'Callaghan has said that he expects the party's five dealbreakers to come into focus when they meet with Micheál Martin and Simon Harris again.
With this in mind, it was decided that talks will also continue with Sinn Féin and Labour.
Labour will meet with Fianna Fáil and Fine Gael on Friday, but the growing expectation is that the party will not go into government.
The Social Democrats negotiating team, made up of Mr O'Callaghan, Jennifer Whitmore, and Gary Gannon, had met with the Fianna Fáil and Fine Gael ministers on Wednesday and updated the party on Thursday.
"We reported that the meetings had been positive and constructive with a good exchange of views," Mr O'Callaghan said.
He said there were detailed discussions with the two leaders around climate, housing, disability, healthcare, and childcare.
The Social Democrats outlined its red line issues during the campaign and said it intends to stand by these.
He said there was "some interest" shown in the proposal to have a senior minister for disability.
"I wouldn't say it was rejected outright or anything like that but it was very initial discussions that we were having in those areas," Mr O'Callaghan said.
The most challenging topic of the discussions was in terms of climate, he said, but added there were further challenges in other areas as well.