The man, who later lost an arm in a road crash, is named in Missing Pieces, by Ian Wishart.
Scheduled for release on Friday, it claims to contain information that will raise new questions about Tamihere's conviction in 1990 for killing Heidi Paakkonen and Urban Hoglin.
Key police evidence and secret witnesses were later discredited, giving rise to discontent over the soundness of the conviction. But Tamihere stayed in jail until paroled in 2010.
Wishart said the new suspect's family told him he confessed before he died in 2002. The man was a mental health patient who had attacked a priest in Waihi shortly before the murders took place.
He later told his family he had escaped into the bush and spent many weeks travelling rough while moving north to Whitianga.
The man's mental health history showed intense anger and violence, Wishart said, which fitted with autopsy records showing Mr Hoglin's head was nearly severed from his body after death.
"Urban Hoglin's body had been subject to a frenzied attack."
Wishart said he had tracked the killer through the 1990s, including a crash in which he lost an arm.
He said it offered further proof of instability - the suspect had his arm out the driver's window at the time he struck a truck. The other driver believed it was deliberate.
Wishart said that after he advertised the book last week on NewstalkZB, a nurse rang to tell him of a man who had confessed to the murders before dying of renal failure in 2002. The man also had one arm.
The nurse confirmed the name Wishart mentioned. She had contacted police to tell them of the death-bed confession but was told the killer (Tamihere) had already been caught.
"It doesn't prove he did it but it does prove he confessed," Wishart said.
He said it remained possible Tamihere was the murderer. "He was a convicted killer and a rapist."
Former Cabinet minister John Tamihere, the convicted murderer's lawyer brother, said discoveries following the conviction threw huge doubt on his brother's guilt. There were links between the case and others that later become synonymous with doubt.
John Tamihere said the police focus on his brother from the outset had removed the opportunity to study others who should have been investigated.
He said his brother was "a great suspect" but had fallen victim to "the policing methodology in this country".
"They clearly identified a key suspect and determined there would be no others."
A police spokeswoman said: "Mr Wishart has not provided police with the material he has referred to in ... promoting his book."