The conservation group believes Horder House, built between 1958 and 1960 in Steep in the Hampshire countryside, is under threat because the elderly owner has put the building up for sale.
The house was built for Cullinan's uncle, Mervyn Horder, as a studio for composing and playing music and entertaining guests.
C20 caseworker Cordula Zeidler said it was extremely important that the threat to the building is relieved. 'We understand the current owner makes it very clear to prospective buyers that demolition is an option,' she said in a letter to the Department for Culture, Media and Sport's listing department.
'We would like to urge you to proceed as fast as possible with this case,' Zeidler added.