The National Trust is closing the islands from Sunday, July 3.
The disease, which was first discovered in UK domestic bird populations last winter, is now impacting wild birds, with infection proving to be fatal.
It is spread when birds come into direct contact with an infected bird, faeces, body fluids or indirectly via food and water.
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The risk to people is considered to be very low and people are rarely affected. Symptoms are similar to normal flu, with a runny nose, sore throat and fever.
Simon Lee, general manager for the Farne Islands, said: “While we have no confirmed test results from the islands, we are now starting to see the terribly sad and distressing impact of Avian Influenza on our internationally important and threatened seabirds who make the islands their home.
“Seabirds nesting in dense colonies, most of which are threatened, such as Arctic Terns, are particularly vulnerable now as they have returned to the islands in their thousands to breed, nesting in close proximity to each other.
“Our ranger teams work tirelessly to monitor and protect these colonies, but due to finding significant numbers of dead birds, we simply have no other choice but to close the islands.
“We understand how many people love to visit the islands, but we must do everything we can to protect and to try to help these much-loved seabirds by limiting the spread of the disease.
“The effect of the disease on the colonies we care for could be devastating due to many species having low reproduction rates, which means the loss of adult birds has a huge impact on populations being able to recover.
Mr Lee continued: “Many of the birds which nest here, such as the vulnerable Atlantic puffin, are already experiencing huge pressures due to climate change with warming sea temperatures impacting food stocks.
“By closing the islands we will reduce the risk of disturbance on the birds, which will hopefully help at least slow down the spread of the disease during this breeding period before they leave the islands in late summer to continue their annual migratory cycle.
The Farne Islands are home to approximately 200,000 seabirds including guillemots, kittiwakes, razorbills and shags in addition to Arctic terns and puffins.
Several hundred dead birds have so far been found with these deaths reported to Defra.
Ben McCarthy, head of nature conservation and restoration ecology at the National Trust, added: “We are committed to working with Government agencies including the Joint Nature Conservation Committee (JNCC) to protect our wild birds, and we are also working alongside other organisations including the Royal Society for the Protection of Birds (RSPB) and British Trust for Ornithology (BTO).
“However, time is of the essence. We desperately need Government to do more to recognise the impact on our wild bird populations and to take more effective action, including a National Response Plan that includes practical steps to support our conservation teams, surveillance and monitoring of spread in wild bird populations, research into ways of slowing its spread, and clear biosecurity measures.”
The ranger team will remain on the islands to continue to monitor the birds.
Boat tours are set to continue with visitors able to sail around the islands.
Mr Lee said: “Inner Farne and Staple are closed to visitors.
"Golden Gate, which operate on a lease to Trinity House still have permission to land at the lighthouse because bird populations on Longstone are very small and outside of the public area accessible to Golden Gate. We will monitor as ever but as public aren’t walking through birds and large populations we feel the risks are reduced at the current time.”
The UK Health Security Agency (UKHSA) has advised that the risk to the general public's health is very low, but people should not touch sick or dead birds. If found, please report any dead birds to Defra on 0345 9335577.