Several incidents were reported in Canonbie and Eaglesfield at the weekend and are being linked by police.
Sgt Steven Wilson, of Police Scotland, said: “There have been some sneak-in thefts — a number of them — and suspicious activity.”
At Forgeholm in Canonbie on Saturday a resident saw a man in his garden walking away from the back door.
Sgt Wilson said: “When he called out to him he ran off and out of sight.
“The person is described as being 5ft 10″ with short cropped black hair, in his early 20s, slim build, wearing a black jacket and brown jumper, and he had some sort of mark on the side of his face.”
In Forgeholm on the same day a resident received a call from a neighbour who had spotted a man on the path leading to the back door.
Sgt Wilson said: “The resident checked her property and discovered it had been entered and a drawer had been opened.”
Purses including one containing about £1000 were stolen, with a similar description of the suspect.
Between 12.15 pm and 1 pm a woman in Aldery Terrace in Canonbie heard her front door being opened and saw a hooded man in his early 20s looking through its window. He ran off to the rear of the property.
During enquiries, a resident of Rowanburn reported a tin containing loose change had been taken from his house that day — later discovered in the ornamental garden on the main road.
And at 12.35 pm last Saturday a Rowanburn resident returned from visiting her neighbour to discover an intruder.
Sgt Wilson said: “As she walked through the back door she observed a man standing in her house.
“The man apologised for scaring her, saying he was looking for directions to Carlisle.”
Ushered out empty-handed, and last seen walking to the B6357, he is described as 5ft tall, of average build, in his 20s or early 30s, short dark brown hair, with a beard, wearing a black puffer jacket and speaking with an English accent.
Jewellery and two laptops were meanwhile stolen from a house in Eaglesfield on Friday January 5.
Appealing for information, Sgt Wilson said: “It seems likely that the same person or persons may have been involved in this type of activity.”