Justine Damond, 40, from Sydney, is thought to have lived with her fiance in the US city.
According to the Star Tribune, her family said she had called emergency services after hearing “a sound” in the alley near her home late on Saturday night.
Witnesses said she was in the alley, wearing pyjamas, when officers arrived in a squad car.
She is said to have gone to the driver’s-side door and talked to the officers through the window.
The police officer in the passenger seat is reported to have pulled out his gun and shot at her.
No weapon was found at the scene and the investigation is expected to include a focus on why the officers’ bodycams were not recording.
The state Bureau of Criminal Apprehension confirmed that an officer had fired a weapon, fatally striking a woman.
Mayor of Minneapolis Betsy Hodges said she was “heartsick” over Ms Damond’s death.
Neighbours of the health and life coach have held a vigil for her in the alley where she died, and her family and friends have described her as “treasured and loved”.
Ms Damond, whose maiden name is believed to be Ruszczyk and who was a qualified veterinary surgeon but working as a spiritual healer, was due to marry 50-year-old Don Damond, whose name she had already taken, next month.
He is reported to have been away on a business trip when she died.
Both officers involved in the shooting, who have not been identified, have reportedly been placed on administrative leave.