Government officials refused to comment on the claim, which could not be independently verified, but officials had earlier ruled out sabotage as a cause of the fire at Istanbul's Ataturk International Airport.
A police official, speaking on condition of anonymity, said an investigation to determine the cause of the fire was still under way.
The Kurdistan Freedom Falcons Organisation, a hardline group linked to the main Kurdish rebel group, the Kurdistan Workers' Party, or PKK, said it started the blaze, which destroyed much of the cargo terminal at the airport, according to the Netherlands-based Firat News Agency's website.
Firat, which often receives information from Kurdish rebel leaders, said it received the claim by email.
The hardline group has claimed eight bombings in Istanbul this year and recently said tourism and economic targets were among the group's priorities. Previous bombings left two dead and 47 injured.
Local television footage showed black clouds of smoke and flames shooting more than 30 metres into the air while firefighting planes could be see flying over the airport.
Three people suffered from smoke inhalation but there have been no other casualties, officials said.
General director of state airport authority, Mahmut Tekin, said the fire had been brought under control and that "there is no problem with air traffic but some flights are delayed".
Istanbul governor Muammer Guler told Turkish television: "There has been no loss of life, but we have suffered large-scale material damage."
Cargo worker Omer Toplar said: "A huge black cloud came, it smelled like cables burning, the roof started burning. There was no explosion."
The cargo building is used to store international cargo arriving in Istanbul. A hangar housing military aircraft is nearby.
The fire caused panic at the airport, as authorities tried to evacuate thousands of people from nearby terminals.