Fianna Fáil has suggested the appointment would breach its confidence and supply arrangement with Fine Gael
An independent member of Cabinet has called for a review of plans to make the former Attorney-General a judge, just as the Taoiseach defended it.
Transport Minister Shane Ross says the nomination of Máire Whelan should be reviewed 'to reassure people' of its validity.
The comments come as Leo Varadkar stands by the appointment, describing Ms Whelan as "the stand-out person for the vacancy".
Fianna Fáil says going ahead with the appointment would breach its Government confidence and supply deal, and potentially collapse the Government.
This is a claim denied by Mr Varadkar, who insists the correct procedures were followed.
Fianna Fáil says it has serious concerns that the role was not contested and that Ms Whelan did not leave the room while Cabinet was considering her nomination.
Sinn Féin deputy leader, Mary Lou McDonald, says there is no way the appointment should be allowed to go ahead.
"It is widely seen as the latest episode of stroke politics,of nod and wink politics - that is the last thing we need on the watch of a new Taoiseach.
"I'd be asking Leo Varadkar as a matter of urgency to put a stop to this particular appointment, to wind back the clock and to act in a manner that is appropriate".
Earlier, a Government spokesman confirmed it will press ahead with nominating Ms Whelan despite the mounting controversy.
Ms Whelan was appointed to the role during Enda Kenny's final Cabinet meeting as Taoiseach.
The role carries a salary of €180,000 a year.
Her name was put forward by the former justice minister, Frances Fitzgerald, after the Judicial Appointments Advisory Board was unable to suggest any candidates.
In a statement, Mr Varadkar said Ms Whelan is an "outstanding legal mind".
He said: "I am very pleased that nobody appears to be questioning her competence or her quality. The issue which is attracting comment however is solely a matter of procedure.
"I am satisfied that the correct procedures were followed. Under Article 13.2 of the Constitution the Government and only the Government can appoint Judges. The Tánaiste recommended Maire Whelan to Cabinet as the stand-out person for the vacancy."
He added: "I am committed to fully respecting the Confidence & Supply Agreement between Fianna Fáil and Fine Gael. I read the Agreement again this morning and it does not require the Government to run public appointments by Fianna Fáil, nor could it."
It comes after Fianna Fáil indicated that a general election could be called unless Ms Whelan either stands aside or is appointed by a fair process.
The party's justice spokesperson Jim O'Callaghan suggested: "[A general election] is a nuclear option [...] but every option will be kept on the table."
Fianna Fáil TD Timmy Dooley earlier said Mr Varadkar has questions to answer - even though he was not officially in the role of Taoiseach at the time of the appointment.
Deputy Dooley observed: "I think it's certainly not acceptable [...] for Taoiseach Varadkar to sort of indicate it happened 'not on his watch'.
"Clearly both Minister Frances Fitzgerald [...] and the incoming Taoiseach Varadkar have very considerable questions to answer here."
Labour leader Brendan Howlin said he 'knows the ability' of Ms Whelan - but raised objections over the process.
Speaking on The Pat Kenny Show earlier, Deputy Howlin argued: "I think she would meet any objective test for appointment to that position. But that's the difficulty.
"On the face of what we know, there seems to have been no objective test and no opportunity for anybody else to apply for it. And that's just not right."
Social Democrat co-leader Róisín Shortall, meanwhile, said: “Mr Varadkar must come before the Dáil next week and give a full account of this murky affair, which totally undermines public trust in our political system.
"If the new Taoiseach genuinely believes his manifesto statement that Fine Gael is a party of integrity and open politics, then he must do the right thing and put the appointment on hold until we know all the facts."