Justine Damond, 40, reportedly called police to report a sexual assault before an officer fired his gun from the passenger seat of the squad car.
Ms Damond, a meditation teacher from Sydney, was standing outside the driver’s door at the time, reportedly dressed in her pyjamas.
She died from a gunshot wound to the abdomen, according to the Hennepin County Medical Examiner.
Authorities in Minneapolis, where the shooting occurred, have not given any details about what happened.
The two officers’ body cams were not switched on and footage from the police car failed to capture the incident.
Don Damond, who is Ms Damond’s fiance, said the family had almost no information about what had happened.
He said: “We’ve lost the dearest of people and we’re desperate for information.
“Piecing together Justine’s last moments before the homicide would be a small comfort as we grieve this tragedy.”
Ms Damond’s family in Australia said they were “trying to come to terms with this tragedy and to understand why this has happened”.
Police Chief Janee Harteau said that she has “many of the same questions” as Ms Damond’s family.
She said: “I’ve asked for the investigation to be expedited to provide transparency and to answer as many questions as quickly as we can.”
Hennepin County Attorney Mike Freeman will decide whether the officers should be charged in relation to the shooting after an investigation.
Mr Freeman has not commented on the case except to say that the officers should have switched on their cameras when Ms Damond approached them.
The officer who fired at Ms Damond was Mohamed Noor, a Somali-American who joined the police department in March 2015.
His lawyer Tom Plunkett confirmed his identity and said in a statement that Mr Noor “keeps (Ms Damond’s family) in his daily thoughts and prayers”.
“He joined the police force to serve the community and to protect the people he serves.
“Officer Noor is a caring person with a family he loves and he empathises with the loss others are experiencing.”
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 Mr Damond, whose name she had already taken, next month.
He is reported to have been away on a business trip when she died.