The Wiltshire town has been among England's worst-affected in recent weeks, with more than 1,020 cases seeing it placed on the government's coronavirus watchlist.
On Sunday, Public Health England reported 24 new cases in the town in a single day. The recent rise in infections has been linked to an outbreak at an Iceland distribution centre.
The Tesco Extra store in question is on Ocotal Way. A spokesperson for Tesco said all the infected staff were self-isolating and the store remained open to customers.
“We have introduced extensive measures across all of our stores to help keep everyone safe, including protective screens at every checkout, social distancing signage and regular deep cleaning,” they added.
Last week, 51 workers at the Iceland distribution centre in Swindon tested positive. The council announced on 5 August the outbreak there is now under control and the number of new cases in the town continues to fall.
A spokesperson for Swindon Borough Council’s Public Health department said: “We are aware of the very small number of Covid-19 cases affecting staff at the Tesco store.
“Our Public Health team is working closely with Tesco and colleagues in Public Health England. All staff affected are self-isolating in accordance with government guidance.”
Steve Maddern, director of public health for Swindon, told Sky News that “small outbreaks” are to be expected as lockdown rules are relaxed and people return to the workplace.
“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,” he added.