The hawk is being sought by WildRescue, a Monterey-based group that specializes rescuing wildlife from danger, director Rebecca Dmytryk said.
Dmytryk said the nail gun shooting could have been accidental but was likely intentional.
Authorities do not know exactly when the shooting occurred, but Dmytryk said her organization received a photo of the hawk on Oct. 9, so the animal has been dealing with the wound for at least 10 days.
She said that although the hawk is in a weakened state, witnesses reported seeing it eat a gopher on Wednesday.
Dmytryk said time is of the essence in finding the bird because "an animal with injuries that is under stress can become compromised" and die of other medical problems such as a fungal infection of the lungs.
WildRescue has been on site in Golden Gate Park for the past three days trying to catch the bird. Rescuers are trying to avoid using nets, which could harm the hawk further, and are hoping to lure it into a trap set up in the park.
The group is short-staffed today and will not be canvassing the park, but will return Friday to look for the bird and post flyers advertising a $10,000 reward for information leading to the arrest and conviction of the person or people responsible for the shooting.
Dmytryk encouraged anyone who sees the bird to take pictures of it and call the group's 24-hour hotline at (866) WILD-911.
She said that with the pictures the group has received, "we're building a history about the bird ... building his pattern of behavior, and can strategize a capture plan."