TheChemist.co.uk says more needs to be done to reassure the population.
The company has seen a high demand for the drug, but was turning people away if they didn't have a prescription even before the government imposed a ban on private sale of Tamiflu and Relenza.
The pharmacy has seen a 700 per cent upsurge in enquiries for Tamiflu between Monday and Thursday this week.
On Monday less than one-in-30 of all the company's enquiries was about medication for swine flu, which had gone up to one-in-five by Thursday morning.
Managing director Mike Rudin said: "The number of calls has been increasing rapidly as the week has progressed, but even before the Department of Health imposed restrictions on the sale of antivirals on Wednesday the majority of enquiries were from people with no flu symptoms and no prescription.
"If any unscrupulous company attempts to sell consumers Tamiflu we would urge people not to buy it, and never use a provider which isn't registered with the Pharmaceutical Society of Great Britain."
Mike Rudin also urged calm on the issue, citing the low numbers of reported infections so far in the UK.
"The government says it has enough antivirals stockpiled to treat half the population and has ordered enough to treat over 80 per cent of it, so we would urge people not to panic and to seek advice from the Department of Health website if they have any concerns," he said.
"However, more could be done to reassure the population about the current risks posed by swine flu, as the level of concern we are seeing is out of all proportion with the current reality in this country."
Health secretary Alan Johnson told MPs earlier this week that the UK has been preparing for a flu pandemic for five years and that all NHS organisations have pandemic plans in place to ensure that antivirals can be made available to the public very rapidly if it should become necessary.