The giant 185,000 square-metre facility near Livingston, West Lothian, is the size of 26 football pitches and supplies over 300 Tesco stores across Scotland, Northern Ireland and the north of England.
Tesco did not confirm the exact number of workers who had confirmed Covid-19 infections but said it was only a “small number”.
They are all now self-isolating at home and anyone they had been in close contact with in recent weeks had been informed about the positive tests.
Despite the outbreak the distribution centre has not closed down and staff are continuing to work two metres apart and wearing masks, gloves and visors, Tesco said in a statement.
“The small number of colleagues that have tested positive for Covid-19 at our Livingstone Distribution Centre are self-isolating and their close contacts have been informed.
“We take the health and safety of our colleagues extremely seriously and are working closely with public health authorities. We have measures in place to ensure colleagues stay two metres apart throughout our centres and hand-sanitiser, masks, gloves and visors available to all colleagues.”
In 2016 the plant, which is just off the M8 motorway, was reported to have 1,800 workers and was open 364 days a year, only closing on Christmas Day.
About 3.6m boxes of food and other produce pass through the facility on their way to Tesco supermarkets across Britain every week.
Food distribution centres and similar warehouses have seen a number of coronavirus outbreaks since the pandemic began.
Food producer Greencore has been forced to temporarily close a plant in Northampton after 292 of its employees caught the virus.
The firm is the world’s largest maker of sandwiches, producing 350m every year, leading some shops to experience shortages.
There have also been outbreaks in poultry factories and other similar meatpacking facilities.
Experts say indoor areas which are cold and damp, such as refrigerated food processing plants, are perfect environments for Covid-19 to spread.
It can also be hard to keep workers separated thanks to fast-moving production lines, while the noisy machinery means staff have to shout to be heard when talking to each other, risking more droplet spread.
However, the Food Standards Agency has reassured consumers it is very unlikely to catch the virus from food, and cooking items will kill it regardless.