It is feared that supplies could dry up as soon as the middle of next week because of problems with the supply chain of the surgical equipment.
Official advice states that a disposable mask must be worn during procedures, so patients could go untreated if the scarcity reaches critical levels.
According to the British Dental Association, more than 50 per cent of masks used by dentists and other medical professionals come from Chinese manufacturers.
However since the Covid-19 crisis started there, supply is said to have “essentially ceased”.
More than 64,000 people have been diagnosed with the coronavirus, almost 1,400 people have died.
Suppliers in the UK have put restrictions on the amount of masks that can be ordered.
Some sellers have run out of stock and others have pushed prices up by as much as 200 per cent.
A number of dentists and health professionals have also been panic-buying as they anticipate a shortage, and the BDA said it has been “bombarded” with calls.
The BDA warns that UK suppliers have issued blanket restrictions, which say dental practices can order no more than two boxes of masks per day – 100 masks in total.
The masks are used by dentists and dental nurses, and are also used when sterilising equipment.
The BDA estimates that a single surgery in a typical NHS practice, seeing around 28 patients per day, will be getting through five boxes of masks a week. They account for about a fifth of all practices in the UK.
But “two-chair” NHS practices are likely to use up their allocation completely, and larger practices may not have enough to continue opening.
Under current guidance, all dental professionals operating in England are told to wear personal protective equipment (PPE) including disposable face masks, clinical gloves, disposable aprons and eye protection.
BDA chairman Mick Armstrong said: “In recent weeks dentists have been hit by panic-buying, clumsy rationing and naked profiteering.
“Sadly a ‘one size fits all’ approach from suppliers is leaving many larger practices with few options.
“Our abiding interest is the safety of our patients, who face imminent disruption to their care.
“Unless we see a rapid increase in supply, dentists without face masks will have little choice but to down drills.”
The BDA said similar issues are being seen in New Zealand, Australia and Canada.
A message from the Australian Dental Association to the BDA said: “Here in Australia, we are potentially going to be facing a situation of mask shortages as our suppliers all rely heavily on Chinese manufacturers.”
It asks what steps can be taken to ensure continued supply and what the BDA is planning to do in the event of masks running out.
A Department of Health and Social Care spokesman said: “The best way to protect ourselves from infections like coronavirus is to wash our hands frequently with soap and water or use a sanitiser gel, as well as always carrying tissues and using them to catch coughs and sneezes, then putting the tissue in a bin.
“We have central stockpiles of a range of medical products, including face masks, to mitigate supply problems and help ensure the uninterrupted supply to the NHS.
“We have well-established procedures to deal with supply problems, regardless of the cause, and work closely with industry, the NHS and others in the supply chain to help prevent shortages and to ensure that the risks to patients are minimised.”