The cases have been confirmed at Palethorpes in Market Drayton, part of the Addo Food Group.
The company is working with Shropshire Council and Public Health England.
It is understood that 38 of the site's 850 workers are self-isolating after 16 of them tested positive.
Rachel Robinson, Shropshire’s director of public health, said: “We are aware that a small number of positive cases of Covid-19 have been confirmed at the Palethropes bakery in Market Drayton.
"We are working closely with the business and Public Health England to help limit any further potential spread. All positive cases are self-isolating for at least 10 days, with contacts isolating for 14 days, as per national guidance."
A spokesman for the Addo Food Group said: “Unfortunately, we have had a handful of colleagues test positive for Covid-19 at our Palethorpes Bakery. We believe that the virus was contracted outside of our facility and these are isolated incidents.
“The health, safety and wellbeing of our workforce is our number one priority - we have implemented and strictly followed government guidance on screening, social distancing and handwashing and continue to take additional steps to prevent further spread of the virus and reduce risk across the business.
"We continue to self-isolate and refer our employees for testing, including those who may be a potential contact of a positive case. As part of our procedure, individuals must self-isolate until a negative test result has been confirmed – it is only at that point that they can then return into the bakery.
'We are now at a critical point'
“Additional safety procedures were introduced into all our facilities at the start of the pandemic and we have made no change to these processes, even as lockdown eased over the last few months. These processes include: two-metre social distancing which covers all work and break areas; safety screens at all working areas; stringent handwashing and hygiene procedures that were in place before the pandemic; and regular cleaning of high touch point areas.
"We continue to follow guidance and review our measures regularly to ensure the safety of our colleagues and minimise the risk of infection."
Ms Robinson warned: “We are now at a critical point in our battle against Covid-19. We are on the brink of having further restrictions being placed upon us here in Shropshire.
"That’s why we are urging everyone to follow the measures designed to keep everyone safe – particularly around observing the rule of six; limiting your social contacts; observing social distancing; frequent hand washing; and wearing face coverings in enclosed public areas, including public transport.”