Justice Minister Charles Flanagan says he would be happy to meet members of the family of murdered Clodagh Hawe to discuss their concerns.
As well as a new and full inquiry, the family have also called for a number of measures to protect other women who may be vulnerable and in danger.
The minister has stated, however, that any proposals would have to be considered by criminal law experts in his department and by the Attorney General.
Alan Hawe killed his wife, Clodagh and their sons, Liam (13), Niall (11) and Ryan (6) in August 2016 in Virginia, Co Cavan, and then took his own life.
Four weeks ago, Clodagh's mother, Mary Coll and sister, Jacqueline Connolly, were refused a request for copies of Garda files into the investigation into the murders.
In a statement, Mr Flanagan said he wanted to assure Clodagh's family that he was totally committed to doing everything he could to address and prevent the crime of domestic violence.
The minister pointed out that An Garda Síochána was currently reviewing cases of domestic homicide to better inform its approach to domestic violence generally.
Clodagh's family want all information gathered in the course of an investigation shared as soon as practicable with the next of kin.
The family, who highlighted their plight on RTÉ television on Monday night, only received the murder/suicide letter penned by Alan Hawe 16 months after he committed the awful crimes.
They want the book of evidence to be published after an inquest into a familicide and family annihilation and that Tusla, the Child and Family Protection Agency, monitors all such cases and maintains research on familicide perpetrators.
They have also called on Minister Flanagan to amend the Succession Act because it makes the perpetrator, or if deceased their family, a beneficiary of the estate.
Mr Flanagan stated that his officials were examining proposals to amend the Succession Act.
“I want to reassure Clodagh's family that we will always do what we can to listen to the concerns of our citizens who find themselves impacted so dreadfully by awful crimes like these,” he said.
The National Women's Council of Ireland has backed calls by Clodagh's family for a multi-agency review into her death and the death of her sons.
The NWCI has also called on Mr Flanagan and Garda Commissioner Drew Harris to complete and publish the findings of the ten-year domestic homicide review that was promised to the council and Women's Aid after Clodagh's death.
NWCI director, Orla O'Connor, said the interviews with Clodagh's mother and sister on RTÉ underlined the importance of multi-agency domestic homicide reviews.
“In other jurisdictions, domestic homicide reviews are systematic multi-agency reviews which are used to improve risk assessment and management and to identify gaps in policy and practice, following such crimes,” she said.
“Ireland must initiate the practice of multi-agency reviews of domestic homicides in Ireland every time such a crime occurs.”
Women's Aid - 1800-341900.
