Officers were called to the allotments on Park Lodge Lane, on the Eastmoor estate, on Wednesday at around 5.50pm, saying that a man was seen trying to take down the rare, protected species.
They spoke with allotment holders and say they are now following lines of enquiries.
The report has also been passed to the local wildlife officer.
The bird is one of just two adult peregrine falcons in the city which nest on the side of Wakefield Cathedral.
A project was launched five years by the Wakefield Naturalists’ Society after the species was spotted at the city landmark.
Since then a live HD camera has been set up inside a specially-adapted birdbox that was placed on the cathedral spire in January 2014.
People from over 100 countries have kept a close eye on the live footage recently after three chicks hatched just five weeks ago.
Francis Hickenbottom, who leads the project, was left angry that the somebody would attempt to kill the birds.
He said: “These people know what these breeds are and I’m sure they know they are protected.
“A lot of people dislike predators, like sparrow hawks or otters, and our peregrines get a lot of attention because of the cameras.
“There are a lot of people in Wakefield who ask about these peregrines who will be unhappy and disappointed that somebody is trying to do this.”
Peregrine falcons are classed as Schedule One under the Wildlife and Countryside Act, meaning they are highly protected.
Anyone caught trying to harm such birds could face unlimited fines.
Many peregrines died out in the countryside in the 1960s and 70s because of pesticide poisoning, but the population is recovering with many nesting in towns and cities.
Mr Hickenbottom added: “It’s going very well because over to the west of Wakefield, in the Dark Peak, no peregrines bread at all last year because of persecution, so the birds we have in Wakefield are in a privileged position.”
Anyone with information about the shooting incident is being asked to contact police on 101 quoting log number 1480 of June 13.