The owl was killed in May, but the appeal has just been issued.
A police spokesman said that the shooting, about 11.45am on Wednesday May 31, was witnessed and enquiries were taking place.
“The person responsible is described as being small or medium build, driving a black 4x4 type vehicle with a dark canopy on it,” said the police spokesman.
“The vehicle thereafter drove off to the B7040 Elvanfoot Road. Witnesses are sought, and those with information are urged to call Police Scotland 101 or Crimestoppers on 0800 555 111.”
This is the second incident alleged to have happened that month. Police have already appealed for information on the shooting of a female hen harrier, killed at 5.15pm on Friday May 4 near the B7040 from Leadhills to Elvanfoot.