Australian Immigration Minister Scott Morrison has said that a question of "moral responsibility" should determine the fate of a baby with Down syndrome, reportedly abandoned by an Australian couple with its surrogate mother in Thailand.

Mr Morrison's comments came as funds raised online by an Australian charity to pay for the infant's medical care rose above €130,000.

More than 4,500 people made donations.

The boy, Gammy, and a twin sister were born to Thai woman Pattaramon Chanbua in December.

She was reportedly paid AU$16,000 (€11,000) to be a surrogate.

An unnamed Australian couple took the sister, who was healthy, but left Gammy behind, according to media reports.

The baby boy also suffers from a life-threatening heart condition and 21-year-old Ms Pattaramon had earlier said she could not afford to pay for the medical treatment he needs.

Mr Morrison said: "I note there was a comment earlier that the mother of this child, baby Gammy, wants the child to remain in Thailand and that mother's wishes also have to be absolutely respected."

Ms Pattaramon told Fairfax Media she wanted to take care of the boy in Thailand, saying: "I'll take care of Gammy on my own. I'll not give my baby to anybody.

"I wish they will love my baby ... I forgive them for everything. That is the best thing I can do, forgive ... it is best for everybody," she said of the Australian couple.

"I want to see all my children back together again. I don't really think too much about the Australian couple. I can't blame them ... I don't feel upset or angry about them anymore. They might have their own problems too."

Mr Morrison said the surrogacy case had some "serious issues" that needed to be managed very carefully.

"But this whole issue I think is fraught with all of these difficulties and I can understand the longing and anguish of parents in this situation who want to be parents, but equally there are some serious issues here that have to be managed very carefully," he said.

Fairfax Media earlier reported that a surrogacy agent who helped coordinate the agreement between Ms Pattaramon and the Australian couple told her to abort the pregnancy after doctors learned one of the twins had Down syndrome.

Ms Pattaramon told the newspaper she refused the abortion because of her Buddhist faith.

Commercial surrogacy, in which a woman is paid to carry a child, is not permitted in Australia.

Couples are able to use an altruistic surrogate who receives no payment beyond medical and other reasonable expenses.

However, Surrogacy Australia said more couples choose to go overseas than find an altruistic surrogate at home.

It said 400 or 500 couples go each year to India, Thailand, the United States and other places to do so.