Shoppers are limited to buying no more than five of certain goods, including antibacterial gels, wipes and sprays, dry pasta, UHT milk and some tinned vegetables
The rules apply in stores and online.
A government spokesperson said it was in touch with UK supermarkets to "discuss their response" to the virus.
Waitrose has introduced a temporary cap on some items on its website, including some anti-bacterial soaps and wipes.
The supermarket said it was in talks with its suppliers to ensure customer demand was met.
It said some individual stores may have introduced their own restrictions, with "some branch managers making a judgement at a localised level".
The High Street chemist Boots has restricted sales of hand sanitisers to two per person.
Asda is also restricting some types of hand sanitiser to two bottles per person - the supermarket's only restriction in place currently.
Meanwhile, Sainsbury's said it was not limiting sales of any products in stores or online yet.
On Monday, Environment Secretary George Eustice is expected to hold talks with supermarket and trade bosses about "support for vulnerable groups who may be in isolation", the Department for Environment, Food and Rural Affairs said.
According to a survey from Retail Economics, as many as one in 10 UK consumers is stockpiling, based on a sample of 2,000 shoppers.
But Dr Andrew Potter, chair in logistics and transport at Cardiff Business School, told the BBC: "Whilst there might be empty shelves at the moment in the shops, over the next week or so, we will see them replenish.
"The supply chain will start to deliver stuff through to the stores and hopefully this shortage - which is fairly short-term - will clear and everything will be back to normal again."
He said while retailers may have been caught out by the beginning of this shopping surge, they had very sophisticated systems to check changes in demand.
Andrew Opie, director of food and sustainability at the British Retail Consortium, echoed that "supply chains remain robust and even where there are challenges, retailers are well-versed in providing measures" to keep shops running smoothly.
Waitrose said it has not put a cap on any of its products in stores.
But it has introduced a temporary cap on certain products on its website, including some anti-bacterial soaps and wipes, "to ensure our customers have access to the products they need".
UK retailers have been warned that they face prosecution if they exploit the coronavirus scare to hike prices for products such as hand sanitisers and face masks.
The Competition and Markets Authority (CMA) has told suppliers to act responsibly and said it was monitoring pricing practices.
It comes as Facebook and Amazon have cracked down on profiteers hiking prices online of face masks and hand sanitisers.
Facebook says it is temporarily banning ads and commercial listings for medical face masks. The ban will also apply to Instagram.
"We're monitoring Covid-19 closely and will make necessary updates to our policies if we see people trying to exploit this public health emergency," Facebook director of product management Rob Leathern tweeted.
"We'll start rolling out this change in the days ahead. We... anticipate profiteers will evolve their approach as we enforce on these ads."
Facebook had earlier announced a ban on ads for medical products which falsely suggested an item was in short supply, as well as those which falsely claimed to provide cures or prevention methods for coronavirus.
Meanwhile, Amazon said it had removed thousands of listings from its sites around the world and was constantly monitoring attempted price-gouging.
Gel price ramping
Analysis from Liberty Marketing has found UK own-brand hand sanitisers are being sold on eBay for huge mark ups, with Lidl 49p sanitisers selling for as much as £24.99 online.
Morrison's £2 hand sanitiser is being sold for £29.99.
The Tesco Health Antibacterial Hand Gel (50ml) is just 75p in-store and is being sold for as much as £9 on eBay.
Other supermarkets included in the research include Asda, with a 2,629% increase, and Morrisons, with a 1,400% increase.
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