A spokeswoman for Boots, the UK's largest chemist chain, said it had seen an increase in the sale of hand sanitisers, but it still had stock available in its warehouses for online sale and in stores.
She also confirmed that there is currently a limit of two hand sanitisers per customer to ensure as many people as possible have access to the products.
A source told the PA news agency that Boots is concerned some of its products could be re-sold online.
The company has seen protective products of the kind it sells, for example face masks, appear on websites such as Amazon at prices far greater than those retailed by Boots.
A spokesman for LloydsPharmacy, which runs over 1,500 UK pharmacies, said it too was limiting the sale of hand gel products to two per person both online and in store.
"We know that having access to products like hand gels is extremely important to our customers, so we are doing everything we can to ensure availability, despite increasing demand and supply challenges," he said.
A spokesman for Well Pharmacy, which has more than 700 branches, said that despite a surge in demand for sanitising products it was currently "not limiting stocks to our customers" and working to maintain stock levels.
"This may result in stock being temporarily unavailable whilst it is replenished from our distribution centre," he added.
"We certainly have no intention of profiteering over the current situation by increasing prices."
On the Boots website, 250ml bottles of Carex hand wash can be bought for £1 each, while packs of six of the product are being advertised for in excess of £16 on Amazon.
A 100ml bottle of Cuticura hand gel is sold for £1.55 by Boots online, while on Amazon sellers are trying to fetch prices of £10 for a 50ml bottle.
On Twitter, users have shared photos of empty shelves in Boots branches as well as images revealing a shortage of hygiene products in some supermarkets.
They also posted photos of signs in Boots stores warning customers of the limit on hand sanitiser.
In the House of Commons on Tuesday, Health Secretary Matt Hancock was questioned on dwindling supplies of hand sanitising gel and whether the UK will have enough supplies of paracetamol.
He said: "Our no-deal planning and our no-deal stockpiles are playing an important part in making sure we are fully prepared and ready."
Sales of hand sanitiser in supermarkets more than tripled last month as worried customers flocked to protect themselves from coronavirus.
Sales soared by 255% during February, according to new data from Kantar Worldpanel.
Meanwhile, liquid soap sales increased 7% and household cleaning products rose 10%.
Andrew Opie, director of food and sustainability at the British Retail Consortium, said: "Sales of hand sanitiser have risen as individuals and businesses make sensible precautions to reduce the spread of coronavirus.
"Retailers are also taking necessary steps to meet the rise in demand for certain hygiene products."