It comes after local media reported he had served time for molesting two girls under 10 in the late 1990s.
The man and his wife took home only one baby from Thailand after the surrogate had twins, leaving behind son Gammy.
The case has made international headlines, causing uproar in Australia.
State authorities told the BBC they were now conducting a "full investigation" to assess the father's "suitability" to have a young child in his custody.
A spokesman for Western Australia's department for child protection and family support told the BBC: "Last night we were made aware of certain information by the police regarding allegations of the father's criminal background."
Asked if convicted sex offenders have been allowed to keep their children, the spokesman said they assess cases based on "individual circumstances".
Besides Down's syndrome, the six-month-old baby has a congenital heart condition and a lung infection.
Surrogate mother Pattharamon Chanbua has been looking after Gammy as well as two children of her own.
She claims his parents abandoned Gammy and had asked her to have an abortion when she was told of the child's condition four months after becoming pregnant.
Ms Chanbua, 21, has said the father met the twins, but only took care of the girl and refused to carry or look at Gammy even though the babies were side by side.
The parents have told local media in Australia that they did not know of his existence, and claimed that the allegations made by Ms Chanbua are lies.
But one local newspaper quoted a family friend saying the parents did know about the boy being born, apparently contradicting their version of events.
"Gammy was very sick when he was born and the biological parents were told he would not survive and he had a day, at best, to live and to say goodbye," the friend said.
She suggested Ms Chanbua, 21, had broken the surrogacy agreement by giving birth in a smaller hospital instead of an international one, which meant that the biological parents had no legal rights to the babies.
The couple had been locked in a legal battle with Ms Chanbua to take home their daughter and she had insisted on keeping Gammy to give him a Thai funeral, the friend alleged.
Both the Australian government and Thai health authorities are now looking into the case and the larger issue of commercial surrogacy in Thailand, which is mostly unregulated.
An online fundraising campaign so far has raised tens of thousands of pounds to help Ms Chanbua with Gammy's medical expenses.
Child protection services in Australia are investigating a man accused of abandoning a baby with Down's syndrome to a surrogate mother in Thailand over allegations of child sex offences.