Refugee and human rights groups have called on the Minister for Justice to heed a recommendation by the Human Rights Commission that non-national parents of Irish-born children not be deported.
A letter to Mr McDowell was signed by groups including Amnesty International, the Immigrant Council of Ireland, the Irish Refugee Council, the National Women's Council. the AkiDwA African Women's Network and the Irish Council for Civil Liberties.
They want the Minister to adopt a recommendation that thousands of families who have applied for residency on the basis of having an Irish-born child not be deported on the back of a Supreme Court decision that they do not have an automatic right to stay in the State.
The letter says such deportations would effectively mean applying the law retrospectively to these people. In some cases, they had withdrawn from the asylum process because they had been advised by lawyers and government officials that they would be entitled to stay on the grounds of having Irish children, the letter adds.
It says granting the families residency would not affect the overall asylum process as they are "a closed group" readily identifiable by the Department of Justice.
It goes on to note that more than 50,000 asylum seekers in Britain, who had waited over three years for their cases to be decided, were recently granted residency to end their uncertainty.
"For families of Irish children, whether former asylum seekers or not, their limbo and uncertainty, too, should end. In this way the families, as well as their Irish children, would be treated fairly and with dignity, not criminalised."
Hundreds of letters have already been sent out by the Department of Justice to the families affected by the ruling. The letters give them 15 days to explain why they should not legally be deported. A small number of deportations has already taken place, but up to 11,000 people may be affected by the Supreme Court ruling.