The local authority has accused Biggin Hill Airport Limited of demolishing the Dispersal Hut at the airfield during Armistice Week last November.
The hut is believed to have been used by World War Two pilots during the Battle of Britain.
According to the airport, sections of the hut were to be “sympathetically restored” and displayed at RAF Biggin Hill Museum and Chapel, Biggin Hill Heritage Hangar and a non-profit heritage foundation in the US.
Bromley Council has said that once it was aware, it acted to “prevent the export of some remaining parts of the building” to the US.
It claims that dismantling the hut was a “serious breach” of the airport’s lease.
The airport says it will now rebuild the hut using materials from the dismantled structure.
“This work means that the hut will remain in its previous airside location at London Biggin Hill, but will not be publicly accessible due to its positioning between two hangars on an operational airport,” a spokesperson for the airport said.
The dispute is now subject to legal proceedings at the High Court.
A council spokesperson said: “The council took legal action following the demolition of a building of historic importance by our tenant.
“The airport is home to invaluable heritage connected to the Battle of Britain.
“When it quickly became clear that the historic building was no longer in situ and parts were being prepared for export to the USA, the council had to act quickly to protect the position.”
The airport said: “Legal proceedings could have easily been avoided or ended at any time by the council if they had responded to any of the airport’s proposals for resolution over the past months.
“Amicable resolution has always been the airport’s preference.
“We remain deeply disappointed by the conduct of Bromley Council and the needless waste of money and resources required by both parties for these legal proceedings.”