A forensic search was carried out at a property in Cessnock, Glasgow.
Mother-of-four Ms Reilly was murdered and dismembered by Andrew Wallace, who was last month jailed for life.
He preyed on the vulnerable 47-year-old, moving into her Govan flat before stabbing her in the chest following an argument.
Her family has since pleaded with the killer to reveal the location of her remains so they can lay her to rest properly.
A Police Scotland statement said: “On March 6, detectives investigating the murder of Julie Reilly attended an address in Lorne Street, Cessnock, where a forensic search of a back garden was carried out.
“During the search human remains were recovered.
“The family of Julie Reilly have been made aware of the discovery.
“A post-mortem examination will take place today and further tests will be required to confirm formal identification.”
Wallace, 42, will serve at least 28 years behind bars for Ms Reilly’s murder.
He admitted the crime and disposing of her body, with her femur bones previously found at two separate locations in the city.
After his sentencing on 8 February, her sister Lynne Bryce appealed to the killer: “Please tell us where Julie is. Let my mum have her wee girl back. Let her first grandchild have somewhere to go and grieve for Julie.”
She added: “I hope that if he has a heart he could now please tell us where he put my sister and let her come home to be put to rest properly.”
High Court judge Lady Rae asked Wallace’s solicitor if he would co-operate with police to recover the rest of Ms Reilly’s remains.
Ian Duguid QC said the killer was “prepared to entertain an approach from the police” once the case was over.
Ms Reilly had suffered a brain injury in 2016 that left her with memory problems and slow reactions.
She was reported missing in February last year after she failed to pick up her prescription at the chemist and her family became concerned.
Two months later, a member of the public called police to say he could see what appeared to be a bone in his front garden in Ardshiel Road, Govan.
Wallace had previously been jailed for culpable homicide and had other convictions for violence.