Shelves have been stripped of essential items, such as toilet rolls, hand sanitiser, paracetamol, meat, fruit and vegetables as shoppers ignore pleas not to stockpile.
It has led to supermarkets having to bring in limits on the amount of some items sold, with golden shopping hours introduced to help the elderly and NHS and care workers.
At a briefing on Saturday, Environment Secretary George Eustice urged people to "be responsible" when shopping and to "think of others."
"Buying more than you need means others may be left without," he said.
Mr Eustice said that it was for retailers to agree what the appropriate limit was on goods such as toilet paper.
"All of the major retailers are working together and exercising their own judgment on where it's appropriate to put limits - item limits - on certain issues," he said.
"Toilet roll is one, where, for reasons that are not really known, there was a spike early on, despite the fact that toilet roll is made in this country and they are able to expand production very quickly.
"That is an item where, to make sure the goods stayed on the shelves, the supermarkets took the decision of putting an item limit."
He added that the Government was keeping "a close eye" on profiteering amid concerns some products have been sold online for inflated prices.
Stores are also taking on thousands of temporary and permanent workers to deal with the increased demand from the Covid-19 crisis.
It is understood Mr Johnson will be speaking to the leading supermarket chains to see what the Government can do to ensure the shelves remain stocked and the supply chains can cope with the demand.
The Road Haulage Association has also welcomed an announcement by the Department of Transport to relax the working hours for drivers for a month from March 23 until April 21.
Tesco, Marks & Spencer, Sainsbury's and Asda have announced a golden shopping hour for NHS and social care workers, so they can join older and vulnerable shoppers in having less competition for restocked shelves.
It comes after critical care nurse Dawn Bilbrough, from York, made a heartfelt plea for shoppers to stop stockpiling, in a video which circulated on social media on Thursday.
In the video, she is seen crying after visiting a supermarket following a 48-hour hospital shift to find there were no fruit or vegetables.
Stephen Powis, national medical director at NHS England, said people should be "ashamed" of themselves for panic buying, and urged people to think of struggling NHS staff.
"I would like to make a plea on behalf of all my colleagues in the NHS, nurses, doctors, paramedics and many, many others who are working incredibly hard at the moment to manage this outbreak of coronavirus," he said.
"It's incredibly important that they too have access to food, to those essential supplies that they need.
"Frankly we should all be ashamed that that has to happen - it's unacceptable. These are the very people that we all need to look after perhaps us or our loved ones in the weeks to come."
Sainsbury's said from next week health and social care staff will be able to shop between 8am and 9am every Monday, Wednesday and Friday, alongside elderly and vulnerable shoppers.
It is also consolidating its opening hours in its main stores from 8am to 8pm, Monday to Saturday, in order to be able focus on restocking shelves. Sunday opening, Sainsbury's Local and petrol station opening times will stay the same.
Chief executive Mike Coupe urged customers to take simple measures to reduce risk by standing one metre away from each other and consider paying with card instead of cash.
"Please also treat our colleagues and other customers with kindness and respect," he said.
"These are unprecedented circumstances and our colleagues are being asked to come to work every day while so many others are being asked to stay at home.
"We all need them to keep coming to work to feed the nation - a small thank you goes a really long way."
Meanwhile, Morrisons is taking on up to 500 staff from Marie Curie and CLIC Sargent charity shops to help the elderly and vulnerable in its supermarkets.
They will be working alongside Morrisons' army of community champions who currently work with local charities and community groups.
The Co-op is donating £1.5 million of essential food items to charity FareShare's network of food banks and community groups, while Asda is giving £5 million to FareShare and the Trussell Trust.
The supermarkets have also begun rewarding staff for working through the crisis.
Tesco has introduced a 10% bonus for its staff paid hourly while Asda is giving employees an extra week's pay.
And to thank NHS workers and community groups for their hard work, Lidl is giving away thousands of bunches of Mother's Day flowers.
It is hoped that easing the restrictions on the haulage industry will help to keep supplies moving and the supermarkets fully stocked as the panic buying shows no signs of waning.
Road Haulage Association chief executive Richard Burnett said: "This is a blanket relaxation covering all sectors and recognises how integrated and inter-dependent supply chains are across the whole economy. The sector is working as efficiently and as quickly possible.
"This relaxation improves resilience in a way that ensure all goods can reach the area where they are needed.
"Shortages are not the problem at the moment - the problem lies with supplying the current excess demand for goods caused by panic buying. This just creates bottlenecks that undermine efficient delivery schedules.
"The relaxation in hours will not reduce the levels of enforcement of the drivers' hours.
"It is vital that companies only use these relaxed rules when needed and companies must monitor drivers to ensure they do not drive tired or in any way unfit. This relaxation must be used wisely, not abused recklessly."
Mr Burnett said the relaxation was introduced following a meeting between the RHA and Transport Secretary Grant Shapps.
Lorries and trailers will also be exempted from MOT testing for three months.
Most vehicles due for testing in the suspension period will be issued with exemptions automatically but some, such as those due for their first test and vehicles returning to service, will require an application.