The incidents have occurred in Wales and come just a day after West Mercia Police revealed officers are trying to trace a dog owner after as many as 28 sheep were killed.
In a tweet, the Rural Crime Team at Dyfed-Powys Police said: "Several sheep have been found recently with severe bite type injuries, upon the Great Forest East Common land, above Trecastle, Brecon.
"These horrific incidents can be avoided. We urge everyone to keep dogs on a lead at all times when livestock are present."
On Tuesday West Mercia Police released an image of a dog after dozens of sheep were killed by what officers suspect are the same animals.
The incidents occurred in Herefordshire, with the first taking place at Upper Gate House in Dinedor at about 9am on May 30.
Then at around 6am on 1 June, it was reported that two dogs entered the same field and struck again. It is believed one of the dogs was the same dog from the first incident.
Some time between 10pm on May 31 and 6am on June 1, a sheep farmer at Lower Huntless Farm in Twyford – approximately one mile away from the other two incidents – reported that his flock had also been attacked by a dog.
The dog was not seen but the injuries sustained by the sheep along with confirmation by a vet confirmed that these had been caused by a dog.
“These are very concerning incidents which are causing a lot of anguish and financial damage for the victims and we are working closely with the farming community to help them maximise security for their flocks,” said Sergeant Mark Jones, of West Mercia Police.
“We can only roughly estimate how many sheep have been killed based on what farmers have told us – we are aware of separate incidents of five, 11 and 12 sheep being killed at different farms so it seems to be as many as 28.
“We have a picture of one of the offending dogs and would ask that the owner or anyone who knows who the owner is to contact me on email@example.com or call me on 07773 034427. The call can be done anonymously.”
The Country Land and Business Association has urged dog owners to keep their pets under close control around livestock.
Incidents of livestock worrying are on the rise and the CLA Midlands team is concerned about the safety of farm animals, particularly lambing sheep.
Stress caused by fleeing from dogs can cause pregnant ewes to abort lambs, injury to the animal and in some cases the death of the animal, which can occur days after the attack. Livestock worrying is a criminal offence.
The branch, which supports farmers, landowners and rural businesses in Shropshire, Staffordshire and Worcestershire, is offering advice to dog owners to help avoid issues.
It is urging people to stick to the Dog Walking Code as well as the Countryside Code.
Owners are advised to keep their dogs on a lead or under close control when walking through or near fields of livestock.
They should also stick to public rights of way and be aware of any livestock grazing in fields that they may have to cross.
The CLA says all instances of livestock worrying should be reported to the police, which helps them keep a true record of the cases and allows them to tackle the issue effectively.