The leaders of the Dáil's political parties and groupings look set to meet with Taoiseach Micheál Martin and Tánaiste Simon Harris on Tuesday to try to agree on a compromise on the speaking rights row.
Labour leader Ivana Bacik said she contacted Mr Martin about arranging the meeting on Tuesday and said the Taoiseach was amenable to the plan.
Independents who have agreed to support the Government want to form a "technical group" which would afford them speaking time and other rights from the opposition benches.
Opposition parties are furious and say the independents want all the benefits of being part of the Government and all the benefits of opposition.
Government TDs are likely to try to push through plans to allow members of the Regional Independent Technical Group to form a "hybrid" group across both Government and opposition at Tuesday's Dáil reform committee.
Speaking to RTÉ News, Ms Bacik said things had become "very difficult" but the opposition wanted to "resolve matters" before the Dáil returns on Wednesday.
Ms Bacik said the opposition was "very concerned there has been some falling back from the Government commitment that an amendment to [Dáil] Standing Orders would be found which was agreeable to both sides".
Despite the criticism, she repeated that a solution could be found: "I think that we can find a resolution of this. I think there can be a sensible meeting of minds on this."
Ms Bacik said this was "simply unacceptable" as it would "make a mockery" of the whole session.
The Labour leader said: "It is a very very important principle that somebody who has negotiated a programme for government as the leader of a group, somebody who has indeed seen their people installed as Ministers within a government, that they couldn't then turn around and be considered as a member in Opposition."
"It smacks of Russia or Belarus to suggest otherwise," she added.
The Tánaiste is said to be "very eager" for the row to be "resolved and resolved quickly", and looks forward to "engaging to help resolve" the matter.
His spokesperson said: "The people of Ireland need a Dáil focusing on the major issues that matter to them and to their community and country.
"In the views of the Tánaiste, this has dragged on far too long."
Opposition not prepared to be pushed around - McDonald
Meanwhile Sinn Féin leader Mary Lou McDonald said opposition parties will consider every option in order to protect their speaking rights in the Dáil.
Asked on RTÉ's This Week programme if that included a walk-out, she replied: "We will consider every single option available to us."
Ms McDonald said that the opposition parties are "not prepared to be pushed around by the Government".
Conflicting interpretations of Dáil 'standing order' rules are at the centre of the political row.
Ms McDonald said that the proposals put forward by the Coalition to try and resolve the row to date are "Alice in Wonderland stuff".
The Dublin Central TD said that the solution was for Michael Lowry and his colleagues to secure speaking time from Government time.
She said that the reality has to be acknowledged that Mr Lowry negotiated the Programme for Government, supports the Government, and his colleagues are Ministers.
Ms McDonald said Opposition parties will not accept any attempt by TDs to "pretend" that they can be in Government and the Opposition at the same time.
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She added that the Leaders' Questions slot was created for the Opposition to hold the Taoiseach and Tánaiste to account and cannot be accessed by Government TDs.
Ms McDonald said it was a "nonsense, a farce and an insult" to suggest that Michael Lowry would put the Taoiseach "through his paces" on Leaders' Questions.
She said that she expected a leaders' meeting to take place next week to try and resolve the speaking time row, although no arrangement has been agreed.
Also speaking on RTÉ's This Week, Fianna Fáil TD Paul McAuliffe said it was not for him or the Opposition to define whether Michael Lowry was a Government or an Opposition TD.
"Both Government and Opposition are faced with a scenario that the current standing rules say very clearly that if you are five Independent TDs and you have no ministers in your group, you are entitled to status of a technical group," Mr McAuliffe said.
