The Minister for Justice, Mr McDowell, has declined to give special consideration to appeals to allow a Nigerian woman, who has suffered a breakdown, to stay in Ireland.
Mrs Elizabeth Onasanwo, who was due to be deported in August with her four children, has been in psychiatric care for the past two months.
Sixteen TDs, including the Labour and Fine Gael leaders, the Union of Students in Ireland and other groups have appealed to the Minister to allow Mrs Onasanwo and her family stay in Ireland on compassionate and humanitarian grounds.
However, the Minister said that every application for leave to remain here on humanitarian grounds was considered carefully on its merits. He made "no exceptions".
"The fact that TDs do or do not weigh in on behalf of individuals isn't the critical factor. The crucial question is maintaining the integrity of our immigration policy and that's what I do."
Ms Rosana Flynn of Residents against Racism said the family was under "inhuman pressure" as a result of their predicament.
"If they are forced to return to Nigeria, this family will be totally destroyed. The psychiatric hospitals there are a travesty."
Mrs Onasanwo's 18-year-old daughter, Christina, has just started college, while she also takes care of her brothers and sisters. Her twin brother, Adewole, is being treated for schizophrenia. Busola, who is 15 years old, attends the Christian Brothers school in Westland Row and six-year-old Bolu is being minded by friends.
The Onasanwo family has lived in Ireland for the past three years. The State rejected their application for asylum on appeal last November, and then refused a request for leave to stay on humanitarian grounds.
However, Garda immigration officers are prevented from executing the deportation order while Mrs Onasanwo is receiving medical treatment. They are also unable to deport her minor children while she is receiving medical treatment.
Mrs Onasanwo said she fled Nigeria because her daughters were at risk of female circumcision ordered by tribal elders. Christina says the family has "lost contact" with their father.
