It is alleged the woman took her two sons to the river at Moama, in the NSW Riverina district, about 5.30pm on Thursday.
She asked the eldest, 9, to get into the water before trying to keep both boys underwater, police allege.
The nine-year-old wriggled free but was attacked by a dog after making his escape and taken to hospital where he remains in a stable condition. The body of his five-year-old brother was found drowned on Saturday morning.
The boys' grieving grandparents have accused child protection services of "miserably" failing the family.
It is understood the concerned couple had been engaging with lawyers for weeks before the tragedy and had raised serious concerns with authorities.
The two boys had been in their grandmother's care but were "snatched" by their mother after her release from jail, their grandparents have claimed.
In a legal letter sent to the Department of Community Corrections, the grandparents warned the boys had been exposed to "violence and physical harm", and said they held the department, and police, responsible, Fairfax Media reports.
It was reported the grandparents' lawyers said the department did not "take steps to protect our client and her grandchildren".
The grandparents had been caring for the boys for close to 18 months, their lawyers said. Their lawyer, Dale Brooks, said the family was devastated.
"They are simply heartbroken," he said on Saturday evening.
Mr Brooks said the grandparents were in a state of shock but rallying behind their older grandson in hospital.
After the grandparents' claims, the NSW Premier said agencies would urgently look into the family circumstances.
Berejiklian said there would be a thorough investigation into what she called "a human tragedy of the highest proportion".