They are accusing the then South Yorkshire coroner, Dr Stefan Popper, of conducting the inquest improperly, and of failing to direct the jurors that they were entitled to conclude the deaths might have been caused or aggravated by 'lack of care'.
The jury at the inquest in Sheffield returned accidental death verdicts by a majority of nine to two.
Last April, the relatives, who are all from Liverpool, were granted leave to appeal - a move unopposed by the coroner, who is now retired, or by the police or South Yorkshire ambulance service.
Lawyers of the relatives argue that restrictions the coroner imposed on the scope of the investigation into how the victims met their deaths meant there was no inquiry into whether or not a better response by the police and emergency services could have saved the lives of the six.
Ninety-five people were crushed to death in the tragedy in April 1989 at the Sheffield Wednesday football ground.