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After workers at the site tested positive for the virus, further testing was carried out.
This led to the identification of more positive results, Leeds City Council said.
Public Health England could not confirm the exact number of cases, but told BBC News the figure was 'less than 20'.
Greggs has said it is working with the local council and PHE to manage the outbreak, having taken 'immediate action'.
The depot is currently undergoing a deep clean, while further contact tracing is also being carried out.
Roger Whiteside, CEO of Greggs, said in a statement: "Following a number of our staff testing positive for COVID-19 at our Leeds distribution depot, we have taken immediate action to implement our COVID response plan and we are working closely with Leeds City Council and Public Health England to ensure that we minimise any possible impact on our customers and the wider community in Leeds and the surrounding area."
Lucy Jackson, Leeds City Council's Deputy Director of Public Health, said: "Following further testing, more staff have been identified as being positive.
"This highlights why further testing and contact tracing is so important which Greggs is proactively undertaking.
"The workplace is being deep cleaned and further contact tracing carried out, with necessary advice shared about self-isolating and awareness of symptoms.
"We are satisfied the necessary steps are being taken quickly to minimise any further spread of infection."
"As the virus is still within our community, we should all continue to keep at least two metres away from people outside our households wherever possible, wash our hands with soap and water often, for at least 20 seconds, avoid touching our face and follow the 'catch it, bin it, kill it' guidance to help keep ourselves and others safe."
Leeds West MP Rachel Reeves said the premises have now closed to be deep cleaned 'up to a Covid-19 safe standard', and will then be inspected by Infection Control and Environmental Health.
Reeves added that she was also waiting for word from Greggs and the council about whether or not workers who are self-isolating will receive full pay.
In a statement posted to Facebook, she said: "I, alongside local councillors from Armley and Bramley Wards, are receiving regular updates from Leeds City Council and are working closely to monitor this situation.
"I would like to thank Leeds City Council for their swift response."
She added: "If you have any symptoms of Covid-19, including a cough, high temperature or loss of taste or smell, you must get a test and self-isolate for ten days.
"I urge people to continue to practice social distancing, wear face coverings unless exempt and use hand sanitiser or wash your hands frequently in order to stay safe."
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