The staff members, who have not been named, worked at food-supplier Cranswick, which is based in Wombwell, Barnsley, Yorkshire.
They were part of the 1,200-strong workforce at the site "maintaining vital supplies of fresh food into the supermarkets", Cranswick said.
In a statement, the company said that the health and safety of staff was its "number one priority".
A company spokesman said their "thoughts and condolences" were with the workers' families, who were being provided with "full support".
The food supplier said that from the start of the pandemic it had taken "additional measures to protect both the physical and mental wellbeing" of staff.
The statement said that "additional cleaning and hygiene measures" had been in place at the site.
It added: "Cranswick employees are designated key workers and are at the forefront of maintaining vital supplies of fresh food into the supermarkets.
"We are doing everything we can to protect them while they carry out this critical role.
"All colleagues are also advised not to attend work if they have any symptoms."
On the company's website it says the site supplies fresh and cooked food products "from farm to fork".