The family found dead in the Republic of Ireland specified the coffins they wanted buried in to an undertaker.
Adrian and Ciara Dunne with Leanne and Shania
The bodies of Adrian and Ciara Dunne and their children, Leanne, 5, and Shania, 3, were found at their home in Monagear, County Wexford, on Monday.
Mr Dunne's mother, Mary, said the undertaker became concerned when Ciara asked for two white coffins.
"He (Adrian) didn't say he was going to do anything," Mrs Dunne told Irish national broadcaster RTE.
"He said the reason he was organising his funeral was because he was afraid
of him having a traffic accident and they would be all killed in it and he
wanted to have everything sorted out before anything would happen.
"He didn't say he had planned to do anything."
The government has said it will set up an independent investigation into the circumstances leading up to the deaths.
Speaking on RTE, Minister for Children Brian Lenihan said the matter had been discussed at length at a cabinet meeting of the Irish government.
Post mortem examinations are being carried out to establish the exact causes of death.
Mr Dunne's brother, James, took his own life a month ago.
Ciara Dunne was originally from Burt in County Donegal.
Gardai broke into the house on Monday and found the bodies of the two girls and Mrs Dunne in the sitting room.
The body of Mr Dunne, who was blind, was found in the hallway.
Police would only confirm that the children did not have significant injuries.
The scene at the family home in Monageer
Gardai said they became concerned for the family last Friday and asked a local priest to visit them. He did so and was concerned by what he found. The following day, officers also made contact with the local Health Service Executive (HSE).
Wexford TD Paul Kehoe has called for a full investigation into the death.
Speaking on BBC NI's Good Morning Ulster on Tuesday, he said: "Anyone that phones an undertaker to organise their burial ground... anyone would know that he was in serious trouble. It is something that time was not going to heal.
"What I am looking for is that Minister (Mary) Harney, through the Department of Health, carries out a full investigation into this."
Stunned relatives and friends comforted each together as the four bodies were removed from the house on Monday night.
Floral wreaths were laid at the scene in the Moine Rua housing estate.