A prisoner has contracted the virus at the all-male category C prison and is in self-isolation, the Express & Star has learned.
Despite confirmation of the first case at the prison, visits are still being allowed.
It is understood measures have been introduced to ensure appropriate distance between inmates and visitors.
G4S, the private company which runs the jail, said it was continuing to monitor national guidance.
John Whitwam, managing director, G4S custodial and detention services, said: “We have confirmed one case of coronavirus at HMP Oakwood.
"The health and safety of our staff and the people in our care is our top priority and we have existing, well-developed policies and procedures in place to manage outbreaks of infectious diseases.
“We are in regular discussion with Public Health England and following their guidance.
"The person concerned had already been self-isolating since Tuesday, in accordance with national guidance, and will remain in isolation and in regular consultation with our on-site health professionals. His family have been contacted and will be kept up-to-date.”
Oakwood joins HMP Manchester and HMP High Down in having confirmed cases of coronavirus.
It is believed visits to Oakwoood have reduced amid the crisis.
The Ministry of Justice said its prisons are primed to take action over the coronavirus outbreak - but that visits are continuing for now.
Officials said "robust contingency plans" were in place for its prisons, including Featherstone, Stafford and Birmingham, but specific measures will not be put in place at individual jails unless evidence of an outbreak becomes clear.
Visits are being allowed to continue, though those with possible coronavirus symptoms are being urged to stay away. Loved ones are been given the option of sending emails or leaving voice messages rather than physically visiting prisons.
The MoJ said measures would include the "isolation of individuals where necessary" but suggested mixing of inmates would not be reduced where there were no known coronavirus cases.
A spokesman said prison conditions were being kept "as normal as possible" but that the situation was "changeable" and Government advice was being monitored.
The spokesman said: "We have put in place robust contingency plans in consultation with Public Health England and the Department of Health and Social Care.
"These use available evidence and prioritise the safety of staff, prisoners and visitors while making sure normal regimes experience the minimum possible disruption.
"As a result of this planning, prisons are well prepared to take immediate action wherever cases or suspected cases are identified, including the isolation of individuals where necessary."