The Opposition and Government parties remain deeply divided over the issue of Dáil speaking rights for some Independent TDs who are supporting the coalition.
A meeting of the Dáil Reform Committee lasting five-and-a-half hours concluded last night without agreement.
The committee - which was tasked with ending the impasse that delayed the election of Taoiseach last week - will meet informally this morning.
But it is not expected that the issue will be resolved until next week at the earliest.
The Government had offered what was presented as a compromise that ensured there would be no reduction in opposition speaking time.
This would have allowed the technical group - made up of Michael Lowry, Danny Healy-Rae and other Independents who were involved in negotiating the Programme for Government - to proceed.
But Dáil rules would change to ensure Leaders' Questions and questions to ministers were extended so as not to eat into the Opposition's time.
In a meeting which was described as tense, the proposal was strongly rejected by the Opposition.
"They were answering a question that we weren't asking," said Labour's Duncan Smith who attended the meeting.
"To us it was about Michael Lowry's group having Leaders' Questions and having priority questions. That's what's unacceptable to us," Mr Smith said.
"They cannot operate as an opposition group after agreeing Programme for Government. That is the principle. We are consistent on that."
Speaking on the way out of the meeting, he said: "I hope there can be some sort of compromise. That will have to come from Government. But I'm deeply disturbed and I'm deeply frustrated after the last five-and-a-half hours of the meeting we just had."
Sinn Féin TD Aengus Ó Snodaigh, who was also in the meeting, said: "The Government still does not understand the fundamental question that there is opposition and there is a Government.
"And in fact they are doubling-down that they want to interfere with the opposition time and interfere with standing orders to facilitate Michael Lowry in particular."
He said it was very frustrating that the meeting went round and round in circles.
"They haven't offered any compromise because they are still talking about Michael Lowry and his team being in opposition even though they negotiated a programme for Government," Mr Ó Snodaigh said.
"I don't know what Michael Lowry has over them that suggests they can't compromise."
Mr Lowry continues to demand that he and three colleagues, who support the Government, are recognised as an Opposition Technical Group for speaking time - along with deputies Mattie McGrath and Carol Nolan.
It is understood that the proposed compromise by Government Chief Whip Mary Butler involves Government deputies being given a slot to raise issues during Leaders' Questions - a key part of the Dáil schedule.
As things stands, only opposition party and group leaders are able to pose queries to the Taoiseach and Tánaiste at midday on Tuesday, Wednesday and Thursday each week.
The deadline set for resolving the issue is today.
Opposition parties have been united in branding the Government compromise text to Standing Orders as unacceptable.
Social Democrats deputy leader Cian O'Callaghan said: "The Government are doubling down on stroke politics."
Sinn Féin leader Mary Lou McDonald described the Government plan as "another stroke being proposed by Fianna Fáil and Fine Gael to allow Michael Lowry to sit on opposition benches".
"You cannot be in Government and Opposition. This is an insulting response to what was agreed in the Dáil last week. Total arrogance on the back of their grubby deal. We won't stand for it," she added.
People Before Profit-Solidarity's Paul Murphy said that Leaders' Questions "is a slot for the opposition to hold the Government to account.
"They want (Michael) Lowry ... to ask them questions about how great they are. Utterly unacceptable."
The draft document is being considered at a meeting of the Dáil Reform Committee.
As the Government has a majority on the committee, it is possible for the changes to be forced through by the coalition.
However, such a scenario would result in the acrimonious row returning to the floor of the Dáil where unprecedented scenes last week delayed Micheál Martin's nomination as Taoiseach by 24 hours.
'Sour grapes' on Opposition's behalf
Independent TD Mattie McGrath said criticisms of the Government's proposal to end the Dáil speaking time row is "sour grapes" on behalf of the Opposition.
Speaking on RTE's Drivetime, Mr McGrath said: "We have a mandate and are we going to be silenced because of this row over."
He said the Opposition is heading towards "another unseemly spectacle next week in the Dáil which is totally unwelcome and not desired by anybody.
"I think it's going to end up here in the full light of day that we'll be calling ourselves reasonable independents instead of the Regional Independents because the Opposition is being unreasonable," he added.
The text of the Government proposal is:
Recognition of unaligned members
Notwithstanding anything in Standing Orders and until further notice in the 34th Dáil, the group of Independent members known as the Regional Group shall be regarded as an informal technical grouping of unaligned members. This group may not include a member who is a Minister or Minister of State.
In this regard Standing Orders shall be amended to allocate additional time for Leaders' Questions and Priority Questions. There will be no reduction in the allocation of time to any party/group. This will provide for the allocation of time to the informal grouping of unaligned members known as the Regional Group. All other speaking time allocations will be considered by the Committee on Standing Orders and Dáil Reform in due course.
Furthermore, notwithstanding anything in Standing Orders, the informal grouping of unaligned members known as the Regional Group shall nominate a member to attend meetings of the Business Committee and the Committee on Standing Orders and Dáil Reform.
Additional reporting Paul Cunningham, Eleanor Burnhill
