Two staff members at Patchway's Asda supermarket have tested positive for coronavirus, Bristol Live understands.
A source close to the 24-hour store on Highwood Lane says there has been a confirmed case in its pharmacy.
The pharmacy worker and one other staff member are understood to have coronavirus. Asda declined to say how many employees are self-isolating.
The source says the pharmacy has been cleaned, but she has concerns over the branch's safety measures.
She added: "The whole store should have been shut and deep-cleaned after positive tests, but it’s been staying open 24 hours a day."
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Asda says "any affected areas" of its stores undergo a deep clean.
The maximum capacity for shoppers in the supermarket has risen from 400 during the first lockdown to around 1,000, the source claims.
"That’s despite there being far more cases in Bristol now than in the first lockdown," she said.
It comes two days after Bristol Live reported claims around 40 employees at Asda's Avonmouth distribution centre have coronavirus symptoms.
Public Health England defines an outbreak as two or more confirmed cases within 14 days of each other.
Asda says the Patchway store has averaged less than 50 per cent of capacity for shoppers over the last week.
An Asda spokesman said: "Since the start of the pandemic we have continuously made our stores a safe place for our customers and colleagues.
"In the event of a positive case at our Patchway store, any colleague who was in close contact will immediately self-isolate as per the track and trace government guidance and be supported with full pay.
"We would like to reassure our customers that along with the regular daily cleaning, any affected areas of the store will undergo a further deep clean to ensure it remains a safe place to shop.”
