One of the men, a 44-year-old, was left with a serious hand injury after the incident in Lund Avenue, Barnsley, which also resulted in a 19-year-old man being bitten on the leg.
The incident happened at 5.15pm on Saturday when the dog was being taken for a walk by its handler. Both bite victims live locally and required medical treatment.
An investigation is being conducted by South Yorkshire Police’s Professional Standards Department, the force said.
It is the latest reported incident in the region involving police dogs biting members of the public.
Last month a woman was attacked by a police dog as she pegged out washing in her back yard in Wincobank, Sheffield.
It came as police, responding to reports of disorder in the street, pursued a group of young men.
Police referred the incident to the Independent Police Complaints Commission.
Yorkshire has the second highest rate of dog bites in England, figures released earlier this year show.
There were 925 hospital admissions for bites in the region in the year to May.
The highest rate of dog attacks took place in Hull and Wakefield.
The data, from the Health and Social Care Information Centre, shows hospital admissions following dog attacks were highest in the North East (21.6 per 100,000 people) and Yorkshire (17.5).
Anyone who witnessed the incident in Barnsley on Saturday should call South Yorkshire Police on 101 quoting incident 886 of September 14.