The case of Essex killer Jeremy Bamber will not be referred to the Court of Appeal, the Criminal Cases Review Commission (CCRC) has said.
Bamber now has several weeks to respond to the CCRC's provisional ruling before a final decision is made.
He was jailed for shooting his adopted parents, his sister and her six-year-old twin sons at their farmhouse in Tolleshunt D'Arcy, Essex in 1985.
Bamber, now 50, was given a whole life sentence for the killings.
He has always protested his innocence and claims his sister Sheila Caffell, who suffered from schizophrenia, shot her family before turning the gun on herself.
A spokesman for the family said they were "very relieved" by the CCRC decision.
'Invited to respond'
A spokesman for Essex Police said: "Essex Police has co-operated throughout with the CCRC in this matter and has no further comment to make at this time."
The CCRC said it had sent Bamber's legal team an 89-page document "setting out in detail the commission's analysis of the case and the reasons for the provisional decision".
"As is usual with commission cases reaching this stage, Mr Bamber and his team have been invited to respond to the commission's case analysis and the reasons for its provisional decision," a spokesman said.
"Given the lengthy and highly complex nature of the case, we have given Mr Bamber and his team three months in which to respond to our provisional decision (usually the period for a case of this type is 40 working days).
"The commission will then consider whatever representations it receives from Mr Bamber and his team before making a final decision on whether or nor to refer the case for a fresh appeal hearing."
Whole life tariff
Last year, it emerged that photographs which were used as prosecution evidence against Bamber were to be reviewed after an expert claimed there were discrepancies on crime scene pictures.
Bamber's defence team claimed police photos of the murder scene could help the mass killer overturn his convictions.
Bamber was given a whole life tariff after being convicted of the murders in October 1986.
In 2009, Bamber lost a Court of Appeal challenge against the order that he must die in jail. He has twice before lost appeals against conviction.