Ashok Kalyanjee, 52, claims there was a failure by psychiatrists to identify that he was suffering from a paranoid personality disorder at the time of the offences.
Kalyanjee could face trial for the fatal assaults on Paul Ross, six, and brother Jay, two, with a jury left to decide if it was murder or the lesser offence of culpable homicide if his appeal succeeds.
His counsel, Gary Allan QC, told judges at the Court of Criminal Appeal in Edinburgh: "The personality disorder was capable of substantially impairing his ability to determine his actions as a normal person would. That failure resulted in a course of action being taken without the benefit of skilled opinion that there may be a basis for a plea of diminished responsibility."
Mr Allan told the court that Kalyanjee pleading guilty to murder resulted in a miscarriage of justice.
Defence lawyers acting for Kalyanjee had originally sought to persuade the appeal judges that the murder conviction should be quashed and a conviction for culpable homicide substituted with a sentence commensurate with such a conviction imposed.
But Mr Allan said he did not think that was appropriate and that at the end of the day the question would be for a jury at a trial.
Three consultant forensic psychiatrists were now satisfied that paranoid personality disorder exists in Kalyanjee's case and he must have been suffering from it at the time of the killings, the defence counsel told the Lord Justice Clerk, Lord Carloway, sitting with Lady Paton and Lady Dorrian.
Kalyanjee's case was referred back to appeal judges by the Scottish Criminal Cases Review Commission which looks into alleged miscarriages of justice.
Kalyanjee collected the boys from their mother in Glasgow and drove them to a lay-by in East Dunbartonshire, where he killed them on May 3 2008. He was jailed for life and ordered to serve a minimum term of 21 years.