Ryan Fischer remains in a stable condition and his injuries are not believed to be life-threatening, after the attack in which Gaga's French bulldogs Koji and Gustav were abducted.
The dogs were handed over to police on Friday.
Now, a source close to the singer has reportedly told The Sunday Mirror that she wants to repay Fischer.
The source told the newspaper: "Gaga feels horrendous about what happened to Ryan and wants to make sure he is looked after.
"She absolutely adores him and is forever indebted to him for his bravery.
"She also wanted to compensate him for lost work, and has been wishing him a speedy recovery."
Gaga - who is currently in Rome working on Ridley Scott film Gucci - had previously offered a $500,000 (£359,000) reward for the safe return of her dogs.
It is unclear whether the woman who handed in the dogs has claimed the reward and her identity is being kept confidential.
Los Angeles Police Department said: "On February 26, 2021 at approximately 6.00pm, a woman met Lady Gaga's staff and Robbery Homicide Detectives at Los Angeles Police Department Olympic Station with the two missing dogs taken in the recent Hollywood street robbery.
"The woman found the dogs and reached out to Lady Gaga's staff to return them. The woman's identity and the location the dogs were found will remain confidential due to the active criminal investigation and for her safety.
"The victim is listed in stable condition after the shooting, and his injuries were described as being non-life-threatening. LAPD Robbery Homicide Detectives will continue to investigate to ensure the persons responsible are ultimately arrested and brought before justice."
Gaga - whose real name is Stefani Germanotta - had previously said that the theft had made her heart 'sick' and that Fischer was a 'hero'.
It is unclear whether Gaga's dogs were targeted specifically, or whether the value of French bulldogs was the motivation behind the dognapping.Featured Image Credit: Instagram/Lady Gaga