Patient representatives met Taoiseach Leo Varadkar and Minister for Health Simon Harris to discuss how the State might formally apologise for overall CervicalCheck failures.
Minister Harris said it was an excellent meeting and an opportunity to catch up on a range of issues regarding the screening programme.
He said he did not want to breach the confidence of what was discussed, but said there would be an apology from the Taoiseach in the next Dáil term.
This was the first meeting to discuss the issue and was attended by Lorraine Walsh and Stephen Teap, who are patient representatives on the CervicalCheck Steering Committee and members of the 221+ support group.
Mr Teap's wife, Irene, died from cervical cancer in 2017.
The 221+ group said the meeting was the beginning of a new dialogue.
It said a full apology was only possible with a full understanding of what people are responsible for.
The group said the discussion was now more properly and fully informed by the work of people such as Dr Gabriel Scally and more recently, Professor Brian MacCraith, which uncovered the shortcomings that led to a catastrophic systems failure.
The group said its objective for an apology was that it begins a healing process for all those affected by CervicalCheck.
"Today's discussions embraced that principle and were approached in a positive and constructive manner, which we hope will now move to a public conclusion in a timely manner," it said.
Recently, the Taoiseach said the nature and format of a formal apology in the Dáil would be discussed with patient representatives.
Last month, he said that he wanted to ensure the apology was adequate and that it would be accepted.
Today's meeting came a day after the publication of the Rapid Review report by Professor MacCraith into CervicalCheck failings.
The Rapid Review was established after RTÉ News revealed on 11 July, that the Health Service Executive and the Department of Health knew then that more than 800 women and many GPs were not told of CervicalCheck results, due to an IT problem.
In the case of around 870 of the women, the problem led to results letters not being issued to them, or their GP in some cases.
In the other 3,200 or so cases, the results were issued to GPs, but not to the women concerned.
The first problem relating to the original CervicalCheck controversy came to light in April last year, as a result of the High Court settlement awarded to Vicky Phelan, who was given incorrect test results and is terminally ill with cervical cancer.
The following month, the Taoiseach apologised to the women affected. The HSE has also apologised for its failings.
It is expected that a State apology will be made in the Dáil at some point after it resumes late next month.