A mother has told a court, "I had to drown my babies" after police alleged she held her two children under the water in the Murray River on the Victoria-New South Wales border.

Key points: Police were alerted when a dog attacked the mother and the older boy in the water

Police were alerted when a dog attacked the mother and the older boy in the water The family are from Deniliquin in NSW

The family are from Deniliquin in NSW The mother turned herself in to Echuca police

The woman was extradited from Victoria to New South Wales on Friday night and charged for attempted murder of one of her sons.

One of the two boys, who are both aged under 10, is in hospital after being injured by a dog reacting to the scene at Moama on Thursday.

His younger brother is still missing, feared drowned, with police divers continuing to search the river yesterday.

The Deniliquin woman, aged in her 20s, faced an extradition order by video-link in a Bendigo court yesterday afternoon, where it was agreed she would be sent to Deniliquin, in NSW, to face a charge of attempted murder of her older son.

Police said the mother took the boys to a property outside Moama, south of Deniliquin, on Thursday at 4:00pm where they went fishing in the river.

Sorry, this video has expired Police Superintendent Paul Condon discusses the charges

She allegedly asked the elder boy to get into the river and dunk his head in the water, the court heard.

It is alleged she then held him under the water in an attempt to drown him.

The court was told the boy was able to wriggle free when his younger brother started screaming on the riverbank and the mother turned around.

The older boy was then "savagely mauled" by a dog.

The ABC understands the dog that bit the boy was reacting to what was happening in the water and was not a family pet.

Victorian Emergency Services personnel search the Murray River. ( AAP Images/Riverine Herald: Luke Hemer )

Police told the court that the woman then held the younger boy under the water.

He was last seen floating in the river and a search of the area continues.

Onlookers came to the older boy's aid, and the court heard they heard the woman say: "I had to drown my babies."

Police said the woman later made full admissions, and told them "I killed my babies".

The older boy was taken to Echuca Hospital before being transported to the Royal Children's Hospital in Melbourne for treatment.

As police read the allegations to the court, the woman occasionally cried, rocked in her seat and held her face in her hands.

Grandmother upset agencies 'fobbed her off'

Police divers searched for the younger boy yesterday.

Lawyer Dale Brooks, who is representing the grandmother of the boys, said the grandmother had approached government agencies several times with concerns about the woman but "they fobbed her off".

"A number of organisations ought to be held to account," Mr Brooks told the ABC.

A helicopter searches for the boy feared drowned in the Murray River. ( ABC News: Karen Percy )

The administrator of Edward River Council in Deniliquin, Ashley Hall, said the community was reeling as news came through.

"This has sent shockwaves, you've got a boy being operated [on] in Melbourne for dog bites and [a young boy] missing, you've got to hope for a miracle but it's not looking too promising at the moment," she said.

The search of the Murray River is being conducted by divers from Victoria Police as well as a NSW search and rescue team, the SES and the Rural Fire Service.

The NSW Police tweeted earlier that the woman had been extradited from Victoria and was being transported to Deniliquin.

She is not expected to be released on bail, they said, and will likely face Deniliquin court today.