The 54-year-old former businessman has twice been found guilty of arranging the murder of his estranged wife Arlene Fraser.
Judges heard an appeal against the second conviction at the Court of Criminal Appeal in Edinburgh today.
They will deliver their judgment at a later date.
Ms Fraser's family, who were in court, said they were optimistic that the appeal would be rejected.
During the hearing, defence QC John Scott argued that a comment made by witness Sandra Stewart was prejudicial and could have caused jurors to do their own research on the case despite having been told not to.
Ms Stewart's disclosure in the witness box last year referred to a previous conviction Fraser had for assaulting his wife to the danger of her life weeks before she disappeared, Mr Scott said.
"It brought into play something that everyone had been trying to keep out of the picture," he told Lord Justice Clerk, Lady Paton and Lord Drummond Young.
The "Google factor" increased the risk of jurors potentially finding prejudicial material on the internet if they searched for it, Mr Scott said.
"If you enter Nat Fraser into a search engine you will get a wealth of material," he said.
"Some of it is accurate, some of it is inaccurate, some of it is innocuous and some of it is prejudicial."
After the jury heard the comment the decision should have been taken to "start again", Mr Scott said.
"There would have been no option but to desert the case."
Mr Scott also criticised media coverage of the case last year before the retrial, branding some of it "outrageous".
One possible outcome of a successful appeal is consideration of a third trial, he said.
Advocate depute Iain McSporran said there was nothing in Ms Stewart's comment to link it to Fraser's assault on his wife.
The jury had no knowledge of the incident, he said.
"There was nothing in her evidence to suggest he had been in prison as a result of being found guilty of anything; nothing in it to suggest it related to anything whatsoever to do with Arlene Fraser," he said.
There was no evidence that jurors ignored the instruction not to do their own research, he said.
The trial judge, Lord Bracadale, dealt with the comment properly at the time, Mr McSporran said.
"It would have been wrong for the trial judge to desert the case," he said.
Fraser was jailed for life and ordered to spend at least 17 years behind bars in May last year for organising the murder of mother-of-two Ms Fraser whose body has never been found.
The ex-fruit and vegetable wholesaler's conviction at the High Court in Edinburgh followed a retrial which lasted five weeks.
The trial heard that Fraser told a former friend that he paid a hitman £15,000 to kill his wife after she began divorce proceedings.
Fraser was found guilty of murder in 2003 but his conviction was quashed in 2011. A fresh trial was granted after the highest court in the land, the Supreme Court in London, ruled that the initial conviction was unsafe.
Fraser has consistently denied involvement in the disappearance of his wife who was 33 when she vanished from her home in New Elgin, Moray, on April 28, 1998.
Speaking after the hearing, Arlene's mother Isabelle Thompson said she was hopeful the appeal would be refused.
"Chances are that they will quash it and that will be the end of it, but you can never be 100% sure," she said.
"We have been through this as many times. Every time it rears its ugly head, you are brought back to the beginning again.
"But we are all coping, we are all trying to get on with our lives and I hope we are succeeding.
"Fingers crossed we get the right decision."
Her father Hector McInnes said: "I was quite surprised that it even got to court.
"I'm optimistic but obviously I'm not a legal professional. I just couldn't see why we had this.
"I'm really quite optimistic that this should be the end of it.
"It would be the end of it for us if he got free, but if we win this particular case, he's still got the right to appeal."
Mr McInnes said Mrs Fraser was always in the thoughts of her family.
"That's what happens when your children go before you. You miss your parents, but you never expect your children to go first."