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Chief executive of Morrisons, David Potts, confirmed that those who fail to follow the rules will be refused entry unless they are exempt for medical reasons.
Potts said: "Those who are offered a face covering and decline to wear one won't be allowed to shop at Morrisons unless they are medically exempt.
"Our store colleagues are working hard to feed you and your family, please be kind."
The news comes after vaccines minister, Nadhim Zahawi raised concerns over people not following lockdown rules during visits to supermarkets.
"I am worried about supermarkets and people actually wearing masks and following the one-way system and making sure when it's at capacity they wait outside the supermarket," he told Sky News.
Medical exemptions for wearing a mask listed on the NHS website includes:
Supermarket chains Tesco and Sainsbury's have said safety remains their "highest priority" and that customers were given regular reminders to follow the rules.
The union Usdaw revealed it has been "inundated" with complaints from its members who say they are "deeply concerned" for their safety as members of the public flout in-store measures.
Usdaw urged supermarkets and food retailers to revert to "stringent" restrictions and said that staff, who are classed as key workers, must be "valued, respected and protected".
General secretary for Usdaw, Paddy Lillis, said: "Retail staff are working with the public every day and not only suffer increased abuse, but are deeply worried about catching Covid-19.
"Where safety measures are agreed, retailers need to make sure that they are being followed consistently, in every store.
"We are also very concerned by reports that too many customers are not following necessary safety measures like social distancing, wearing a face covering and only shopping for essential items.
"These key workers must be valued, respected and protected."
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