The Independent Office for Police Conduct (IOPC) said it had taken a statement from a man alleging he was attacked by the gunman outside a Plymouth supermarket in 2016.
“We are examining what action police took following the report and whether their response was appropriate,” a spokesperson added.
The IOPC did not identify the victim but a man previously told the Mirror that Davison, then 17, had brutally attacked him and his pregnant partner in an unprovoked assault outside a Plymouth Tesco in 2016.
“The police didn’t seem to take it very seriously, they said he was ‘nothing to worry about’,” Scott Hobbs said.
“If they had logged it as a proper assault surely it would have flagged up when he applied for a gun licence. I wish I’d pressed harder for them to do something about it. I feel maybe [the shootings] would not have happened.”
The IOPC is also examining why Devon and Cornwall Police returned Davison's gun and permit to him in July, after it was removed following another allegation of assault in September 2020.
Davison murdered five victims before killing himself on 12 August, in the the worst mass shooting in Britain since 2010.
The 22-year-old shot his mother Maxine Davison, also known as Maxine Chapman, at a house in Biddick Drive before he went into the street and murdered Sophie Martyn, three, and her father, Lee Martyn, 43.
In the 12-minute attack, Davison then killed Stephen Washington, 59, in a nearby park before shooting 66-year-old Kate Shepherd, who later died at Derriford Hospital.
The incident was not declared a terror attack, despite evidence of Davison’s interest in the incel movement online.
Any ideological factors will be examined at inquests into the deaths of the gunman and his victims.
The IOPC said it had submitted a preliminary investigation report with a file of accompanying evidence to the coroner.
Regional director David Ford said: “At the end of our investigation, we will decide whether any individual has a disciplinary case to answer as well as whether there is any wider learning for the force, the police service nationally or other agencies involved with firearms licensing processes.
“We are now working to finalise a substantial report, together with our conclusions, early next year.”
A pre-inquest hearing on Thursday was told that Davison's GP declined to comment on whether he was suitable to hold a firearms licence when asked by the police.
He applied for a shotgun certificate in July 2017 and after the application was processed by Devon and Cornwall Police, a certificate was issued to him in January 2018 which was valid for five years.
A review by University Hospitals Plymouth NHS Trust into Sophie’s death is considering whether Davison’s GP responded appropriately to a request for information from police.
The doctor declined to provide information and said he was not “in a position to be qualified to comment on the personality of a patient to say whether or not they were safe to hold a gun licence”.
Davison had received mental health support during the coronavirus lockdown and had been in contact with a telephone helpline service in Plymouth. A further pre-inquest review will take place on 15 March.
Patrick Maguire, a lawyer representing relatives of Ms Shepherd, Mr Washington and the Martyns, said: “My clients are greatly concerned about how Davison came into possession of the firearm against the backdrop of matters aired at today’s hearing.
“We welcome the IOPC investigation and we will now await its conclusion and full report in order to respond appropriately.”
In the wake of the shootings, the Home Office announced the police will now have to check someone's medical history before issuing a gun licence.
All firearms applications must be accompanied by a medical document signed by a registered, practising doctor.
New statutory guidance also says any relevant health records - particularly any information on mental health, neurological conditions and substance abuse - will have to be reviewed as part of the process.
It means police, for the first time, will be legally required to follow the guidance to help improve standards and consistency across forces in the UK.
Police have also been told to review an applicant's social media accounts and financial history as well as carry out domestic violence checks in cases where officers believe more evidence is needed before authorising a licence.
Additional reporting by PA