A three-month review is to be carried out by Law Reform Commission President Mr Justice John Quirke to recommend the best means of providing support and payments to the survivors.
Survivors of the Magdalene Laundries have been invited to contact the Department of Justice to register their intent to seek State support.
Minister for Justice Alan Shatter said while a simple structure is being devised, people should register their intent by contacting the Department of Justice on 01-476 8649.
The department had received around 200 calls to the number by early this afternoon.
The authors of the McAleese Report estimated that over 800 former residents are still alive.
The Justice for Magdalenes Campaign has urged the Government to give statutory powers to Mr Justice Quirke.
The group said survivors and their families who were in the Dáil yesterday found the Taoiseach's apology a deeply meaningful experience, which they had never anticipated.
People can also contact the department by writing to:
Magdalen Laundry Fund,
C/O Department of Justice and Equality,
Montague Court,
Montague Street,
Dublin 2.
President Michael D Higgins today described the apology to the women made by Enda Kenny yesterday as "very, very generous".
He said: "But what was most important was that the women themselves were very satisfied."
Mr Kenny told the Dáil this morning that one of the strong points that came from the Magdalene women was that they wanted a State apology, but they also wanted an effective, clear process that was non-adversarial and not legally dense.
Speaking during Leaders' Questions, Mr Kenny said Mr Justice Quirke was trying to set up a process that would enshrine those principles.
Meanwhile, survivors of the Protestant-run Bethany Home have called on the Government to determine responsibility for deaths, neglect and abuse in the former Dublin-based institution for unmarried mothers that doubled as a female prison.
Earlier, Minister for Education Ruairi Quinn said he would like to see survivors of the Bethany Homes being extended the same treatment from the State as that given to the Magdalene survivors.
However, speaking on RTÉ’s Morning Edition, Mr Quinn said a decision on the issue did not rest with him.
"I would like to see that they were treated in the same way as the Magdalene people have been treated if that is possible, but I'm not in a position to comment. It would really be up to the Minister for Justice and for Mr Justice Quirke in the first instance," he said.
Minister for Transport Leo Varadkar said he believes the religious orders should make a contribution to the redress scheme for the Magdalene survivors and should also offer an apology for their role.
Speaking on RTÉ's Today with Pat Kenny, the minister said yesterday was one of those days when the Taoiseach spoke for the whole country.
Mr Varadkar said it will be extremely difficult for the Government to find the money for the redress scheme, but it will do it.
He said the money, to a certain extent, will come from borrowings, but the Government must also protect resources for existing child protection.
A survivor of another institution, which was not included in the McAleese report, has welcomed Mr Kenny's State apology.
The Taoiseach included the survivors of Dublin's Stanhope Street Training Centre in his apology in the Dáil last night.
Speaking on RTÉ’s Morning Ireland, Julie McClure said she could not believe the apology they got from Mr Kenny, which went beyond what she expected.
Ms McClure said she thought it was going to be a matter of fact statement and something he had to do, but she said it was a lot more than that.
"To me it felt he was personalising it to each and every one separately," she said.
Ms McClure said she now wants to be paid for the three years she worked at Stanhope, and described herself as a slave there.
She said she was not believed and even her own mother could not visualise the industrial laundry she was working in.
Stanhope was not recorded as a Magdalene Laundry even though many women spent many years there working under very harsh conditions.