Kwasi Kwarteng, the business secretary, said he is in contact with business chiefs and "monitoring the situation" closely.
His comments came after multiple reports of bare shelves in supermarkets across the country, with pictures posted on social media showing supplies of some products completely depleted.
Several of the UK's largest chains including Tesco, Sainsbury's and Waitrose said their supply chains have been hit by staff, such as delivery drivers, having to self-isolate.
Under current rules, a person who is notified by the NHS app that they have come into contact with a Covid-positive case must quarantine - even if they test negative.
Warnings of staffing shortages grew as NHS figures showed 618,903 alerts were sent to users of the coronavirus app in England and Wales in the week to 14 July.
Earlier this week the government said that a number of critical workers, including those working in the health and care sectors, will be exempt from the rules if they have had both vaccines doses, although no formal announcement has been made.
In an interview on Thursday morning, Mr Kwarteng declined to put a date on when the plans would be published, but warned the list of exemptions would be "quite narrow".
He told Sky News: "Obviously you have to draw the line somewhere and what we want to do is to keep people safe and that's why we have the self-isolation policy in the first place."
When pressed on which workers might be on the list, the business secretary said: "I can't tell you, because you'll have to wait and see when we announce the list," adding we "should know very soon".
As pressure mounted on the government throughout the morning Mr Kwarteng later told the BBC the list would be published on Thursday as he urged shoppers not to panic over supermarket shortages.
Several of Thursday's newspapers had pictures of empty supermarket shelves on their front pages.
Richard Walker, Iceland's managing director, said the supermarket was having to hire 2,000 temporary workers to prepare for "the exponential rise in pinging".
But he told BBC Radio 4’s Today programme that the images seen in the media "are isolated and not widespread".
He said ministers - not shoppers - should be the ones "panicking" and called on the government to include retail workers and HGV drivers on to the key worker list.
The British Retail Consortium (BRC) had previously urged ministers to "act fast" to allow fully inoculated workers to skip self-isolation.
Quarantine rules will not change for the rest of the country until after 16 August when most Covid rules have been phased out.
But Helen Dickinson, the BRC's chief executive, warned that not including retail workers and delivery drivers on the exemptions list could result in reduced shopping hours and store closures.
"I think what the most important thing for government to do is to recognise that the current situation is untenable," she said.
Boris Johnson announced the plan for a "small number" of critical workers to be able to continue their roles despite being pinged as he scrapped most of England's Covid restrictions on Monday.
Former health secretary Jeremy Hunt said the government risks "losing social consent" for isolation if it does not immediately bring forward the relaxation of quarantine rules for the fully vaccinated.
On Tuesday the PM's official spokesperson said: "It's not a blanket exemption and my understanding is we're not going to be producing a list covering individual sectors, these business-critical areas will be able to apply for exemptions to their host departments."
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