The borough council said this afternoon that staff at the Iceland distribution centre at XPO Logistics in Bridgemead had been infected.
Three firefighters from Swindon Fire Station have also tested positive, the Dorset and Wiltshire Fire Service confirmed.
The service added that anyone else who develops symptoms will be tested and the firefighters have isolated for 14 days.
Assistant Chief Fire Officer James Mahoney said: “We appreciate that this will be of concern, and I want to reassure people that we have had stringent procedures in place since the start of the pandemic to protect staff and members of the public at incidents, including the wearing of appropriate PPE at all times.
“In line with public health guidance, all staff who have been in close contact with those who have tested positive have been asked to self-isolate for 14 days.
“The Service has invoked its business continuity arrangements to ensure that the outbreak is contained and managed. We have established processes to ensure that our emergency response is maintained at Swindon fire station and elsewhere, and anyone needing the help of the fire and rescue service should be assured that we will still respond to 999 calls with the required resources.”
“Small outbreaks” have also been reported at Honda and Royal Mail’s delivery office in Dorcan this week.
Swindon director of public health Steve Madder said the cases were no cause for alarm.
He said: “Small outbreaks are expected as a result of relaxing lockdown arrangements and as people return to work.
“We have developed a local outbreak management plan to supplement and support the national test and trace service, and to ensure that Swindon can respond swiftly to help manage a local outbreak or hotspot.
“The council is currently responding to a small number of Covid-19 cases, and these people are isolating in accordance with the government’s self-isolation guidance.
“All workplaces remain open and are following the existing cleaning guidance for non-clinical settings advice, including continuing to provide staff with guidance on social distancing and hygiene to limit spread.”
Mike Wade from Public Health England South West said: “We are working closely with Swindon Borough Council and local employers to provide public health advice to stop the virus spreading.
“Everyone should be following social distancing advice, both in and out of work, to prevent the spread of the virus.
“Good hygiene is the best prevention and there are some simple steps you can take to protect you, your family and colleagues by washing your hands regularly and thoroughly and if you cough, cover your mouth and nose with a tissue.”
Councillor Brian Ford, the cabinet member for adults and health, said: “The wellbeing of the local community is always our priority and our thoughts are focused on the recovery of those that have tested positive for Covid-19.
“We are working with a range of organisations to do everything we can to minimise the spread of the virus in Swindon. PHE has issued extensive advice and guidelines to support them in managing confirmed cases of Covid-19.
“We all have a part to play in reducing the risk of infection and must continue to follow social distancing guidance while at work and socialising.”