Named the Dodington Art Gallery, Wilkinson's design will be built among trees in the listed walled garden at the Dyson's secluded Dodington Park estate in Gloucestershire, England.
The gallery will be a discreet addition to the grounds, taking the form of a "low key pavilion", and contain the family's collection of modern art that includes works by the likes of Pablo Picasso, David Hockney and Andy Warhol.
"The Dodington Art Gallery has been carefully designed to create a discreet piece of architecture that fits comfortably into its particular context," explained Wilkinson.
"It is constructed with quality materials that relate to the historic brick walls of the garden and it sits low within the site amongst the trees."
Dodington Art Gallery, which has now been submitted for planning, is set to become the first major modern art gallery in the area and will be free for the public to access on a limited number of days each year.
It will be positioned adjacent to an existing sculpture garden that visitors will also be able to explore, containing a water feature by the Dyson founder and art commissioned by his wife.
Wilkinson's design for the gallery is composed of a series of walls in an angular arrangement, unified by a band of clerestory glazing and a thin overhanging roof.
These walls will be clad with patinated bronze panels, chosen for a contemporary aesthetic that also echoes the ancient red bricks of the historic walled garden.
Steel columns concealed by the walls will support the roof, which will project three metres from the edge of the building. According to Wilkinson, this is designed to partially overhang the existing garden wall and help to exaggerate the "horizontal nature of the building".
The roof will be complete with a carpet of sedum plants, while its diagrid structure will be exposed internally.
Visitors will enter Dodington Art Gallery from the northeast corner of the building, where a glazed pivot door will puncture the garden wall.
Here, they will be greeted by a lobby that provides access to the exhibition spaces where the art will be displayed along the wall.
The lobby will also contain a staircase, passenger lift and goods lift to provide access to a hidden basement gallery. The basement will mirror the plan of the ground floor but also contain bathrooms, a reception and a cloakroom in place of an existing plant room.
All required energy for the gallery is expected to be provided by a ground source heat pump.
The Dyson family is reported to have a deep-rooted appreciation for modern art, with Dyson once aspiring to become an artist himself. His wife, Dierdre, is an established painter and carpet designer.
Among the works in their collection that will be exhibited is a print from Warhol's Toy Series, Crying Girl by Roy Lichtenstein, Hockney's Domestic Interior Scene, Broadchalke, Wiltshire and pieces by Picasso, Yves Klein, Victor Vasarely and Allen Jones.
Architect Wilkinson co-founded architecture studio WilkinsonEyre with Jim Eyre in 1987, with offices in London and Hong Kong.