The Independent Alliance wants the Dáil recalled to debate the issue
The Cabinet has failed to reach a decision on appealing the Apple tax ruling during today's special meeting.
Ministers will meet again on Friday morning.
In a statement, the Government said ministers had a 'thorough' discussion on the decision, and examined the full text of the European Commission ruling ordering Apple to pay €13 billion in back-taxes.
"Following the discussion, it was agreed to adjourn the meeting to allow further time to reflect on the issues and to clarify a number of legal and technical issues with the [Attorney General's] Office and with officials," it adds.
Fine Gael ministers have been clear that the ruling had to be appealed to the European Courts of Justice, and want a clear decision on Ireland's response as quickly as possible.
But the Independent Alliance wants the Dáil recalled to debate the issue.
They also want a strongly-worded motion on tax, in exchange for supporting an appeal.
Minister Shane Ross believes the Cabinet needs more time to examine the document, adding that "no decision will be made until Friday".
Separately, Minister Katherine Zappone welcomed today's adjournment, saying: “Given the complex issues involved it would have been wrong to rush into a decision today.
"I welcome the fact that the Taoiseach and cabinet colleagues have recognised my concerns and are allowing time for the issues to be further explored and addressed."
The Taoiseach Enda Kenny had earlier hinted a decision did not have to be made today on an appeal.
Finance Minister Michael Noonan had been seeking agreement on an appeal at today's special Cabinet meeting.
Labour and Fianna Fáil both indicated their support for an appeal against the judgement.
Sinn Féin, meanwhile, called for a public inquiry into Apple’s "sweetheart deal" and committed to bringing forward a Dáil motion opposing such an appeal.
The Social Democrats, Anti-Austerity Alliance and Green Party also backed the EU ruling.
The CEO of Apple, Tim Cook, said the claim that the company was given a special deal had "no basis in fact or in law".
He called the judgment an effort to "rewrite Apple’s history in Europe, ignore Ireland’s tax laws and upend the international tax system in the process".
Ireland has two months and 10 days to challenge the findings.
The tech giant could be forced to pay Ireland up to €19bn in back taxes to Ireland when interest is taken into account as a result of the decision.
Speaking on his way into Government Buildings, Mr Kenny said everyone needed time to digest the ruling.
Meanwhile, a senior member of Cabinet has said that Ireland must be allowed set its own taxes.
Public Expenditure Minister Paschal Donohoe told Newstalk Breakfast that the Government was not complicit in tax avoidance.
"I'm very clear that the right thing to do here for the country in the long run is to appeal this ruling," he said.
"Any economic activity that Apple undertook in Ireland they were taxed on by our Revenue Commission.
"Ireland has not been complicit in any such arrangement, and similarly it is not our role - nor do we have the ability - to tax economic and commercial activity that doesn't take place in our jurisdiction.
"The effect of the Commission ruling yesterday is they basically want Apple, which as you know is the largest company in the entire world, they want all of their tax on all of their global economic activity to be taxed out of Ireland.
"That is not something that we can do."
Meanwhile, Jobs Minister Mary Mitchell O'Connor insisted Ireland will appeal the ruling.
She said there should be no question marks about Ireland's 12.5% corporate tax rate.
Ahead of this morning's Cabinet meeting, she also said an appeal must and will be lodged.
Communications Minister Denis Naughten said it was a case of Europe attempting to dictate Irish tax law.
Foreign Affairs Minister Charlie Flanagan said he hoped there will be support for an appeal.