Georgia, a black and white Border Collie cross-breed, went missing from the Brownswall estate, in Sedgley, at around 8.15pm on Tuesday.
She had been walking off her lead when she got "spooked" and ran off, said Luke Bradbury, 22, who was out walking his grandmother's dog.
It is believed the dog, which is micro-chipped, was seen around Wood Farm Riding School, off Gospel End Road, Sedgley, on Wednesday.
Extensive searches of the area have been carried out.
Mr Bradbury said: "It was about 7.45pm [on Tuesday] when I took her a walk. She got spooked and basically legged it.
"I believe there might have been a sighting around Wood Farm going towards Baggeridge way.
"We haven't had her long. She is a rescue dog from Romania."
In a Facebook post published by Jill O'Callaghan, Mr Bradbury's cousin, she said Georgia was "very timid" and "has an area of damage around her right paw".
"She is wary of people but friendly and not aggressive in anyway," she said.
K9 Search UK, a dog rescue organisation based in the West Midlands, is helping to find Georgia.
The group has asked people not to catch Georgia, if seen, but instead note the time and date, and location, of when the dog's sighting.
Anyone with information is asked to call 07931393492 or 07988433187 or visit www.k9searchuk.com.