Professor Sir David King made the statement as an increasingly worried public reported thousands of sightings of dead birds over the weekend.
Almost 3,500 people called the Department of Rural Affairs last Friday and Saturday and tensions among the public also increased after details of a leaked government contingency plan warned that a pandemic could kill 100,000 children in the UK.
Sir David, speaking on ITV1's Jonathan Dimbleby programme, said that the Government was preparing for it as a "very low possibility".
He added: "We have got a virus in the bird population that has gone on since 1996, and in Asia particularly there has been a lot of contact between human beings and the birds that have got that virus.
"The one swan doesn't mean it has arrived here. We need to see more evidence of spread before we can say that it has arrived in the UK."
Sir David emphasised that no more wild birds had tested positive for the deadly H5N1 strain since a case was found in a swan in Fife last week and that British poultry had also been unaffected by the virus.
The Sunday Times yesterday reported details of a study by Sir Liam Donaldson, the chief medical officer, who advised the Government that 50,000 child deaths could be prevented by a widespread closure of schools. He said that in his assessment of the consequences of the H5N1 avian flu virus mutating to threaten humans, it would take just a single case to be confirmed in a teacher or pupil for all the schools in a county or borough to be closed.
Another leaked government report included plans to call off-duty firefighters and retired lorry drivers into service to ensure food supplies were delivered.
Documents reportedly outlined concerns about a lack of preparation among food firms, and a potential shortage of HGV drivers who were willing to go into infected areas.
Experts at the Central Science Laboratory in York are continuing DNA tests on the dead swan, which was found washed up in the harbour of the coastal town, in an effort to determine the exact species it belongs to. Since February, 1,100 birds have been tested for avian flu in the UK.
The bird was found in an advanced state of decay, with its head missing, making it hard to identify. Finding out which species the bird belongs to will help epidemiologists know if it was migratory and where it might have come from.
Until that information is available, Defra is refusing to speculate about how the bird may have picked up the disease.
Jack McConnell, Scotland's First Minister, warned against being complacent because other European countries affected by bird flu had seen more than one case.
Mr McConnell said: "That doesn't mean there may not be another case in Scotland or somewhere else in the UK at some time over the next few weeks and months."
So far, more than 100 people worldwide have died of the highly pathogenic H5N1 strain of the disease, but all have been in close proximity to infected poultry.
The H5N1 virus cannot pass easily from one person to another and therefore currently does not pose a large-scale threat to humans.
But experts fear the virus could gain this ability if it mutates. They say it could trigger a flu pandemic in its new form, potentially putting millions of human lives at risk.
A Defra spokesman said: "If in any doubt, people should call the hotline. But the criteria for calling is if people find one dead swan, duck or goose, three dead birds together of the same species or five dead birds of different species."Reuse content