It was constructed in the shape of an H in a nod to Henry VIII and was previously home to one of the infamous Gunpowder Plot conspirators, who planned to blow up the House of Lords on November 5, 1605.
Claudia and Matthew - who have children Caspar, 17, Clementine, 15, and 10-year-old Cosima - filed the planning application in August.
At the time, the pair were also said to be looking into converting a room at the Tudor house into an estate office to improve security.
A report into the work stated: "The alterations proposed are restricted to the ground floor north service wing which is a single storey addition to the main part of the house, enclosed within a walled courtyard.
"The wing presently provides three rooms and a lean-to shed.
"The majority of this accommodation is finished with uninsulated concrete floors, bare brickwork and flint rubble walls and an exposed roof structure with a partially sloping plastered ceiling.
"All these rooms are presently under-used in an important location at the main entrance to the house and the proposed new uses will improve catering facilities, administration and security of the site, with minimal change in the existing appearance or risk of damage to historic fabric."
The showbiz power couple splashed out a reported £7.5 million on the historic mansion in 2002.
At the time, the luxury abode wasn't listed for sale, but the pair rocked up and asked the owners if they would consider parting with the property.
