Hands Across the Water was also arranging on Saturday for the boy, named Gammy, to be transferred from a hospital in Chon Buri to a private hospital in Bangkok, the Sydney Morning Herald reported on Saturday.
The charity was chosen by the organisers of the Hope for Gammy online fund-raising campaign, who have raised A$150,000 (4.5 million baht) as of Saturday to help the boy.
Gammy was born to Pattharamon Chanbua, 21, of Chon Buri in December. The agent who arranged the surrogacy took his healthy twin sister but left him behind.
The unidentified Australian couple have faced a firestorm of criticism for "abandoning" the boy, but it is not clear whether they knew any details about the case, or even that the surrogate was carrying twins.
It has been reported that when the agent who arranged the surrogacy learned that the boy would have health problems, Ms Pattharamon was asked to terminate her pregnancy. She refused on religious grounds.
Gammy was born with Down's syndrome and also has a heart condition that requires surgery. He was admitted to hospital on Thursday and his condition was critical but stable on Saturday. A spokesman for the charity said the child should make a healthy recovery but would face years of medical care.
"I’ll take care of Gammy on my own. I will not give my baby to anybody," Ms Pattharamon told the Sydney Morning Herald.
She expressed gratitude for the funds that had been raised on Gammy's behalf and said she hoped to be able to use some of the money to help other babies with health problems.
"They [the surrogacy agency] told me to carry a baby for a family that does not have children. ... They said it would be a baby in a tube," Ms Pattharamon told the Australian Broadcasting Corporation.
She said she agreed to carry the child for a fee of A$16,000 (475,000 baht) - enough to educate her own two children and pay back debts.
"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,” said Ms Pattharamon.
"I don’t feel upset or angry about them anymore. They might have their own problems too."
Australian Prime Minister Tony Abbott said the Gammy story was an "incredibly sad" situation.
"I guess it illustrates some of the pitfalls involved in this particular business," he told reporters on Friday.
Paying for surrogacy is illegal in Thailand, according to Tares Krassanairawiwong, deputy director-general of the Department of Health Services Support at the Ministry of Public Health.
"Surrogacy can be done in Thailand but it has to comply with the laws," he said. "A surrogate has to be related to the intended parents and no money can be involved."
The Gammy case has raised concerns that Australia will ban international surrogacy.
Commercial surrogacy, in which a woman is paid to carry a child, is not permitted in Australia but 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, with 400 or 500 each year venturing to India, Thailand, the United States and other places to do so.
"It's just much easier overseas," chief executive Rachel Kunde told AFP. "There's so much red tape involved (in Australia)."
Ms Kunde said the details of the latest case were not clear, and it was not known whether the Australian couple involved were even aware the boy had been born or had been told that he had been aborted.
She said that Australia, which has no national legislation on surrogacy, meaning some states already banned commercial surrogacy arrangements overseas, needed better regulation.
"Our greatest fear is that Australia is going to ban international surrogacy altogether," she said. "We are hoping that the government will make accessing surrogates in Australia easier."