My Brother The Serial Killer, being shown today, focuses on Glen Rogers, a carnival worker whom Florida jurors convicted in 1997 of killing a woman in a Tampa motel room.
Rogers, now 50, was also convicted of murder in California and is a suspect in killings in Mississippi, Louisiana and Kentucky and possibly several other states.
Most of his victims were women he had met in bars while drifting across the country and all were stabbed to death.
Rogers was arrested in November 1995 near Waco, Kentucky, after a nationwide manhunt for the so-called Cross-Country Killer and a 100mph chase.
Rogers, from Hamilton, Ohio, met Nicole Brown Simpson in 1994 when he was living in Southern California, his family says in the documentary.
Anthony Meoli, a criminal profiler, says in the film that he received a painting by Rogers of the murder weapon used in the 1994 murders of Ms Simpson and her friend Ronald Goldman.
"I believe that Glen believes he killed them," said Mr Meoli, of Atlanta, who has received more than 1,000 letters from Rogers and has interviewed him in prison.
Ex-football star and actor Simpson was accused of the killings but the so-called "trial of the century" in Los Angeles ended with his acquittal in 1995.
Simpson never gave evidence at the criminal trial, but memorably demonstrated in court that a glove found near the scene did not fit his hand.
He spoke at length in a wrongful-death trial that led a Los Angeles civil court jury in 1997 to find him liable for damages in the case.
Simpson is currently serving a prison sentence in Nevada after being convicted in 2008 of leading five men, including two with guns, in a September 2007 confrontation with two sports memorabilia dealers and a middleman at a Las Vegas casino-hotel.
Much of the documentary is narrated by Rogers' brother Clay, who used to rob homes with Glen Rogers as a teenager but in 1993 called the police on his brother after finding a body at the family's Kentucky cabin.
Clay Rogers said that in 1994, weeks before Nicole Brown Simpson's murder, Glen Rogers told him he had met her.
"'They've got money, they're well off and I'm taking her down'," he quotes Glen Rogers as saying.
Other family members also said Glen Rogers talked about meeting Simpson's ex-wife.
Mr Meoli said Rogers told him that OJ Simpson paid him to break into Nicole Brown Simpson's house to steal a pair of $20,000 earrings.
Other clues, Mr Meoli said, were that Rogers drove a white pick-up for his construction job -- a white pick-up was seen near the Simpson house on the day of the murders -- and a second bloody footprint at the scene was never identified.
Rogers' family also said he sent his mother a gold angel pin with a diamond. Rogers later wrote to Mr Meoli saying that he had sent it to his mother the day after the Simpson murders and implied that he stole it from Nicole Brown Simpson.
"It's something everyone missed," he wrote. Rogers' mother wore the pin at his Florida murder trial.
"All those things put together a plausible alternative theory," Mr Meoli said.
In a statement, Mr Goldman's sister criticised the documentary.
"I am appalled at the level of irresponsibility demonstrated by the network and the producers of this so-called documentary," Kim Goldman said.
"This is the first time we are hearing about this story and considering that their 'main character', Glen Rogers, confessed to stabbing my brother and Nicole to death, you would think we would be in the loop."
The film's director, David Monaghan, said investigators should look into Rogers' claims of killing Ms Simpson and Mr Goldman and the inmate should be held accountable for several other murders around the country as well.
But a district attorney who prosecuted Rogers, and a detective who interviewed him in connection with an unsolved killing said the convicted killer may be lying in a misguided effort to get off death row.
Los Angeles County deputy district attorney Patrick Dixon, who prosecuted Rogers for the September 1995 murder of a woman in the San Fernando Valley, said there was no mention in that trial of the Brown-Goldman killings, which occurred more than a year earlier.
Mr Dixon said Rogers' brother, who sat through the trial and gave evidence in the penalty phase, never mentioned the OJ Simpson case. Clay Rogers later wrote a book and "none of this was in the book", he said.
He said Glen Rogers' modus operandi did not match that of the Simpson-Goldman killer other than that he stabbed his victims.
Asked why Rogers would now claim responsibility for the high-profile case, Mr Dixon said he might be trying to get sent back to California.
"He could be getting close to execution in Florida," he said. "They move faster there."
Dan Frazee, a retired sheriff's deputy who questioned Rogers about a 1992 unsolved murder, said: "He's got nothing to do in prison right now but sit there and play games."
Mr Frazee said Rogers was "the most evil person I've ever talked to".