The two cases of COVID-19 are unrelated and the individuals affected were confirmed to have contracted the virus outside of the borough.
Julia Burrows, director of Public Health at Barnsley Council, said in a statement that the risk of transmission is low, and advised people to go about their daily business as normal.
There is no evidence of the flu-like virus spreading within the town, but Ms Burrows said the department would not be announcing any new cases and asked people 'not to speculate about individuals'.
"We send our best wishes to the individuals and their families as they continue their recovery," she said.
"I want to reassure residents that Public Health England has contacted anyone who has been in close contact with these cases, and we’re working closely with health partners to manage the situation.
"I do want to emphasise to our residents that people should continue to go about their daily lives as normal. The small number of confirmed cases in the borough suggests that we’re still in a period of containment and we’re not yet seeing evidence of it spreading in the community.
"We know it’s a rapidly evolving situation, so I do want to prepare residents for the possibility that we will see an increasing number of cases in Barnsley in the coming weeks or months. On this basis, we do not intend to announce new cases and we would encourage people not to speculate about individuals.
"We’ve been reviewing all of our council business continuity plans to make sure we are in the best possible position to respond."
Ms Burrows said Barnsley residents should follow NHS advice on how to avoid spreading or catching the virus.
- Wash your hands with soap and water often – do this for at least 20 seconds.
- Always wash your hands when you get home or into work.
- Use hand sanitiser gel if soap and water are not available.
- Cover your mouth and nose with a tissue or your sleeve (not your hands) when you cough or sneeze.
- Put used tissues in the bin straight away and wash your hands afterwards.
- Try to avoid close contact with people who are unwell.
- Do not touch your eyes, nose or mouth if your hands are not clean.
Those concerned about symptoms should contact the NHS online coronavirus service or call NHS 111, rather than contacting or visiting a GP or hospital.
You can find more information at www.nhs.uk/coronavirus