Taoiseach Leo Varadkar has described the illegal adoption registration revelations as another dark chapter in the country's history.
He said it will be a very difficult time for many people and the issue must be handled with the utmost sensitivity.
The Taoiseach said it was too early to to talk about things such as DNA testing, and the focus should be on giving people the information they should always have had.
Speaking later in the Dáil, Mr Vardakar said there were potentially "hundreds of thousands" of further illegal adoption registration records in existence.
Responding to Sinn Féin leader Mary Lou McDonald, who asked if the information about St Patrick's Guild that emerged yesterday was "the tip of the iceberg", Mr Varadkar said an initial scoping and sampling exercise would take place to look at records of other adoption societies.
He said if there was evidence, a full analysis of those records would be carried out. "It's potentially a mammoth task", he said.
Labour TD Jan O'Sullivan asked if the Taoiseach would commit a full audit of all the adoption records, as well as wraparound supports by the child and family agency for those who receive information about their adoption.
The Taoiseach said Tusla had notified the Commission of Investigation into Mother and Baby Homes and gardaí have been given ten sample cases.
Mr Varadkar said those operating the adoption agencies may be elderly or deceased, however he did not rule out prosecutions following the garda examination of the records.
"I'm assured Tusla is aware of the sensitivities involved and will ensure the cases progress sensitively," he said.
He also told Ms O'Sullivan that a social worker had been assigned to each of the 126 cases, who will provide counselling and support throughout the process.
Adoption agency issue a failure by the State - Donohoe
Separately, Minister for Finance Paschal Donohoe said the issue of children incorrectly registered was a failure by the State.
Speaking on RTÉ's Morning Ireland, Mr Donohoe said it now falls to the Government to respond to those affected. He said the first thing to do is identify those affected by the controversy.
Mr Donohoe said it was too early to talk about redress and it was more important to inform those identified in the review, and then to see how many more people may be affected.
Meanwhile, the Chief Executive of children's charity Barnardos has said that every adoption agency in the State has been involved in illegal adoptions.
Speaking on the same programme, Fergus Finlay said: "Yes, it is a scandal, but it's not a new one and shouldn't come as a surprise to anyone."
The names of the people they were placed with were incorrectly recorded as their birth parents.
Incorrect or "false" registrations occur where a child is placed with a couple or individual who was not the parent, but the birth is then registered as if the child had been born to that couple or individual.
Mr Finlay said an investigation now needed to be carried out into at least 150,000 adoptions. He added he would be "amazed" if at least 10% of those cases were not found to be illegal.
"It was a crime against the children because it robbed them of their identity, their heritage, their health histories."
He added that it was also a crime against their mothers.
"A lot of women were told after their babies were taken from them that their babies had died and not to inquire about them again. Some women have gone through years and years of trauma," he said.
The Adoption Rights Alliance has said that there will be more people affected by the revelations of falsified birth certificates.
Speaking on RTÉ Six One News, the co-founder of ARA said she hopes that the Government’s proposed scoping and sampling exercise leads to a "comprehensive and complete audit."
Susan Lohan called on the Government to consult with groups such as the ARA to collect intelligence.
She also said she was concerned that Tusla was named as the main player in the scoping exercise, adding that it was not properly resourced.
Today Minister for Children Katherine Zappone contacted politicans who put down amendments in the Seanad to the Adoption Information and Tracing Bill 2016.
The bill focuses on helping people to trace their birth parents; however, the bill has remained at committee stage.
The bill provides that birth parents can put forward compelling reasons not to be contacted.
Legal issues have been cited - balancing the right of the children and the right to privacy of the birth parents.
However, some legal experts believe the legal issues have been exaggerated.
Ms Zappone is seeking to meet senators to resolve the difficulties with the bill.
She has announced that a review of samples of records from other adoption societies to see if there was any evidence of similar incorrect registrations.
The review will be conducted by Marion Reynolds, a former deputy director of social services in Northern Ireland, and will be completed within four months.
Tusla is operating a Lo-CALL number 1890 10 00 54 between 10am and 4pm on weekdays for anyone who may be seeking information and additional information is available on the Tusla website.