Sainsbury's and Morrisons will enforce mask wearing and shopping alone in its stores, the companies have said.
In a statement, Sainsbury's - which has a superstore in Newport - said it has written to customers to ask them to play their part in keeping everyone safe, and said it has "significantly reduced" the number of customers allowed into stores at any one time.
Simon Roberts, Chief Executive of Sainsbury's, said:
"I've spent a lot of time in our stores reviewing the latest situation over the last few days and on behalf of all my colleagues, I am asking our customers to help us keep everyone safe.
"The vast majority of customers are shopping safely, but I have also seen some customers trying to shop without a mask and shopping in larger family groups.
"Please help us to keep all our colleagues and customers safe by always wearing a mask and by shopping alone.
"Everyone's care and consideration matters now more than ever."
Sainsbury's said it would put trained security guards at store entrances to challenge those not wearing a mask and those who were shopping in groups.
Morrisons - which has stores in Newport and Lake - will also be enforcing the measures.
David Potts, Chief Executive of Morrisons, 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."
Vaccines Minister, Nadhim Zahawi, said the government was concerned that supermarkets were becoming places where the virus could be spread more easily.
"We are concerned that, for example, in supermarkets we need to make sure people actually wear masks and follow the one-way rule in supermarkets.
"And, of course, when they are at capacity - to operate safely, people wait outside supermarkets.
"We don't want to go any tougher, because this is a pretty tough lockdown."
There are only a few cases where people can be exempt from wearing face masks in settings such as supermarkets, including children under the age of 11, people who physically cannot put a mask on or wear one, and people for whom putting on, wearing or taking off a mask would cause "severe distress".
Also exempt are people accompanying those who rely on lip reading, or those in cases where using a mask would risk harm or injury to others.