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The biological parents of seven-month-old Gammy are accused of leaving him in Thailand after he and his healthy twin sister were born to a local surrogate.
His sister was brought back to Australia.
The couple, who have not been identified but live in Western Australia, deny the allegations and claim they had no knowledge of a second child, ABC reports.
Gammy's 21-year-old Thai surrogate mother, Pattaramon Chanbua, is now his full-time carer and says those claims are wrong.
"This is not the truth," she told ABC's 7.30 programme.
Ms Chanbua said the Australian couple asked her to abort the boy after he was diagnosed with Down syndrome in the womb.
She cried, refused and told them she would take care of both the babies if they did not want them.
They eventually agreed to take the girl.
Immigration Minister Scott Morrison has labelled Ms Chanbua a "saint" and an "absolute hero" and said the outpouring of support was a strong indicator of the way Australians felt about the situation.
Gammy's story could prompt the Australian government to look closer at surrogacy laws with the Department of Foreign Affairs already examining practices in Thailand.
Agencies were working with Thai officials on the broader surrogacy issues, the department said.
Mr Morrison said the legalities surrounding international surrogacy were "very, very, very murky" and regulations must be looked at carefully.
"Sure, there are lots of Australians who are desperate to be parents but that can never, I think, sanction what we have just seen here," he said.
Prime Minister Tony Abbott has also said the case illustrates the pitfalls of international surrogacy.
Commercial surrogacy, where a woman is paid a fee to carry a child, is illegal in Australia.
However, if there is an agreement for the biological parents to cover just medical and other reasonable costs, the practice is legal.
Gammy's picture has been beamed around the world and has prompted donations of more than $200,000 to pay for his future in Thailand.
"Our hearts are filled with love for all the people around the world who have taken time out of their lives to give thought to Gammy," his donation page reads.
The donations meant Gammy could be moved to a private hospital with specialist care and Ms Chanbua promises all the funds will go to him, ABC reports.
"I will make sure all the money, every single Baht, is well spent for the best of the baby," she said.
She says even if his biological parents come back for him, she's now his mother.
"I will not allow them to come to take my baby away," she said.
"I love him."