The World Health Organization has warned people against panic-buying stocks of a drug they hope will protect them against bird flu.
Romanian authorities are taking no risks with the potentially lethal virus
The WHO stressed that Tamiflu could reduce the effect of the illness but was not a vaccine to prevent it.
A BBC correspondent says the WHO wants governments to stockpile Tamiflu, but fears a public run on the drug.
Meanwhile bird flu's spread across Europe continued with confirmation that it had reached Greece.
The authorities in Romania - where a number of birds have tested positive for the disease - also announced on Tuesday that they had found swans with suspected bird flu near the Ukrainian border.
Scientists have warned that a strain of the disease deadly to humans could cause a lethal pandemic if it mutates into a form that can be spread from person to person.
The H5N1 strain has killed more than 60 people in South East Asia since 2003. However, of those only one is suspected to have died after catching the virus from another human, and experts stress the risk is low.
Following reports that European pharmacists were selling out of Tamiflu, the WHO's pandemic alert chief Michael Ryan said: "WHO at this point does not advise for individuals to stockpile this drug for any purpose.
"There is no indication at this stage for anyone to be taking this drug other than the very high-risk groups in areas in which the avian disease has become a problem," including poultry farmers and medical staff, he said.
The manufacturer of Tamiflu, Swiss drugs company Roche, has said it would donate packs of the anti-influenza drug to Turkey and Romania, where bird flu has appeared, as well as to the WHO, and was stepping up production of the drug.
In Greece, the agriculture ministry said a form of the bird flu virus had been found in a turkey on a farm on the Aegean Sea island of Oinouses.
H5N1 BIRD FLU VIRUS
Principally an avian disease, first seen in humans in Hong Kong, 1997
Almost all human cases thought to be contracted from birds
Possible cases of human-to-human transmission in Hong Kong and Vietnam, but none confirmed
Twelve swans have also tested positive for bird flu in a second cluster in Romania, and the authorities have confirmed the virus has been detected at a third location in the Danube delta.
And the European Commission has ordered urgent tests on dead birds found in Croatia.
At this stage, none of the new outbreaks in Greece or Romania have been confirmed as the lethal H5N1 form of the bird flu virus.
However, tests are continuing, and the Greek outbreak is known to involve the H5 strain, of which the deadly form is a member.
An outbreak of H5N1 was confirmed in Romania last week, and it has also been detected in Turkey.
The H5N1 strain remained largely in South-East Asia until this summer, when Russia and Kazakhstan both reported outbreaks
Scientists fear it may be carried by migrating birds to Europe and Africa but say it is hard to prove a direct link with bird migration