The woman and her friend were walking their dogs in Holbeck Hill, Scarborough, on Monday morning when they were approached, police said.
A dog, thought to be an English bull terrier, attacked their pets and bit one woman’s leg, leaving her wounded on the floor.
Officers want to speak to the dog owner, said to be a man aged 40 to 50.
North Yorkshire Police said the owner was described as about 5ft 9in, white and slim with either very short hair or a bald head.
He was wearing a black waist-length coat with a hood.
The dog which attacked had a tan stripe on its back and a white neck.
Anyone with information about the incident, which took place at about 09:20 BST, is urged to contact North Yorkshire Police.