The former physician was sentenced to four years in jail last November after being found guilty of the 'Billie Jean' singer's involuntary manslaughter.
Murray, currently serving his sentence at the Los Angeles County Jail, is believed to have filed legal documents yesterday (January 9) asking that evidence omitted from the trial be taken into consideration when deciding how much money he will give the Jackson family.
If this proves that Michael administered the fatal dose of propofol himself in June 2009, relatives will see their payment halved due to the shared responsibility of his death, TMZ reports.
The Jackson family are apparently seeking $100m (£64m) in restitution, though a judge has yet to settle on a figure.
On January 4, Murray was given 15 days to hand back his medical certificate.
He is believed to have hired an attorney to lead the appeal against his manslaughter conviction.