Former independent government minister Katherine Zappone approached Minister for Foreign Affairs Simon Coveney to offer herself for a position of special envoy before her appointment was made.
Ms Zappone served as an Minister for Children in the last Fine Gael-led government, but lost her seat in Dublin South-West in the 2020 general election.
Her appointment as special envoy for freedom of expression has been strongly criticised by the Opposition.
It has also caused tension between the coalition parties after the Taoiseach said he was not given advanced notice before it was brought to Cabinet.
Micheál Martin told reporters today: "In relation to the appointment of Katherine Zappone, Minister Coveney has accepted it was an oversight in terms of procedures, he brought the appointment to Cabinet."
The Taoiseach added: "We move on now - it is an issue you have to keep in perspective."
The Green Party's Government spokesperson said it is important that the appointment of envoys is looked at to ensure a more transparent system is in place, and that best practice is followed.
The spokesperson said that both the Tánaiste and the Minister for Foreign Affairs have acknowledged that they should have flagged the appointment well in advance of yesterday's Cabinet meeting and "we are glad that they have done so."
Tánaiste Leo Varadkar has said he or his advisors or Minister Coveney should have informed the Taoiseach ahead of her appointment being brought to Cabinet for approval yesterday.
He said: "The procedure would normally be that - at the advisors' meeting - the names are flagged and in this case they were not.
"In terms of a belt-and-braces approach, the line minister or me as party leader would have flagged it to the other party leaders or to the Taoiseach and it didn't happen and it should have happened and we accept responsibility."
Speaking on RTÉ's Today with Claire Byrne, Mr Varadkar said it is "nonsense" to suggest that her appointment amounted to "cronyism" and that it is "not a plum well-paid position".
He said: "A special envoy position is not one that is typically advertised. It is a position where somebody is approached or offers to do a particular job, usually a retired diplomat or retired diplomat.
"My understanding is that [Katherine Zappone] approached Minister Coveney."
Mr Varadakar added: "She is available, she offered to do the work."
He denied it was a plum position, saying: "She will get around €15,000 before tax and put in hundreds of hours of work for that."
He said Ms Zappone got the job because she is the most qualified person.
"She knows the system.She is based in New York. She has been a national and international leader on human rights, LGBT issues and that is why she offered to do it."
Taoiseach @MichealMartinTD says that he was not notified ahead of Cabinet yesterday, regarding the appointment of Katherine Zappone to new UN role.— Tommy Meskill (@TommyMeskill) July 28, 2021
He says Minister Simon Coveney accepts that it was "an oversight in terms of procedures. We move on, we leave it at that" @rtenews pic.twitter.com/BVyq8C2F9F
Fianna Fáil TD Barry Cowen called for "clarity" on the events that led to Ms Zappone's appointment.
Speaking on RTÉ's Drivetime, he said those in "responsible positions" should "clarify that this is kosher".
Mr Cowen also said he found it "bizarre" for the Taoiseach to say that he was "blindsided" by the appointment and the decision should have been the Cabinet's "collective responsibility".
"I did think it a bit bizarre that the Taoiseach should say that he was blindsided by the proposal. Cabinet decisions are about the collective Cabinet responsibility. You'd expect the Cabinet as a whole to back the proposal, or if not, to block it or at least seek more time or information before making a decision.
"We all need to know what exactly the post is, what are the requirements, what is the workload, what are the terms and conditions, what is the salary, and what are the expectations of the cabinet in relation to that role," he said.
Earlier, Sinn Féin spokesperson on public expenditure and reform Mairéad Farrell criticised the "lack of transparency and due process involved in appointing former Ms Zappone to the taxpayer-funded role of UN special envoy for freedom of expression".
She said: "It is nothing more than Minister Simon Coveney looking after a former government colleague, who was voted out of office at the last general election.
"The only transparency about this process was the see-through faux outrage from Taoiseach Micheál Martin - posturing about having concerns about the appointment while simultaneously rubber-stamping it."
The Rural Independent TDs group described the appointment as a "deeply cynical and arrogant development".
Leader of the group, Tipperary-based Mattie McGrath, said appointing Ms Zappone to the UN delegation "without advertising or interviewing for the role, smacks of political patronage and is an outright abuse of power".
Aontú leader and Meath West TD Peadar Tóibín said: "We are calling on the Government to reverse the decision and create a transparent recruitment process that is open to all those qualified and interested."