Officers are investigating two reports of sheep worrying on the fields at Kirkwood, Dalton, Lockerbie.
In the first incident, which happened back on 21/22 October last year, in a field known locally as Burnies, on the Kirkwood Estate, six sheep were attacked and three subsequently died of their injuries.
In a second incident on the estate, on Wednesday 17 January 2018, on a field known locally as Burn, three sheep were attacked, injured, and subsequently died of their injuries.
Constable Lisa Duff said, "The reporter is an experienced farmer and is sure that on both occasions the sheep were attacked by what he describes as a large dog, or dogs.
"We are keen to hear from anyone who may have any information about these incidents. Again we would want to warn the public about the control of dogs in and around livestock and devastating results that can happen when they are left to roam where livestock are situated.
"Anyone wishing to contact us regarding this or other incidents of livestock worrying should use the 101 number and quote the reference PDG0015420118."