Paltrow and Arts Club chairman Gary Landesberg purchased the 132,000-square-foot (12,300 square metres) plot that was once the flagship of erotic boutique Hustler for the members' club.
The pair have enlisted the help of Gensler's California office to transform the building into a nine-storey club, complete with a wellness centre, restaurants, cinema, nightclub and a subterranean car park.
Unlike it's original London counterpart, the West Hollywood branch will offer some public amenities including retail spaces and an art gallery on the first floor. Offices will occupy floors two to four.
The fifth floor will comprise a private dining terrace, screening room, and gym; the sixth floor will house 15 guest rooms for members.
A restaurant, another terrace and a lounge for members will be located on the seventh floor, while a Japanese–Latin restaurant is planned for the storey above. A further terrace, swimming pool and helipad will be located on the building's roof.
Vertical aluminium fins covering the building's facade will move and pivot upon axial fittings.
The Arts Club was founded in 1863 by authors Charles Dickens and Anthony Trollope, and sculptor Frederic Leighton, in hopes of providing a space for those interested in the arts, literature or sciences.
In 1896, the club relocated from its original home on Hanover Square to its present location inside an 18th century townhouse on Dover Street.
Membership to the Los Angeles Arts Club is expected to cost $3,000 (approximately £2,126) a year.
Paltrow's proposal is still awaiting planning consent for various aspects of its design – notably the structure's nine-storey height – according to Curbed.
It is the latest development slated for Hollywood's Sunset Strip, and will be just a few blocks away from a branch of another members-only club: Soho House.
Last year, architect Frank Gehry unveiled plans for a development of five buildings along LA's famous strip.
According to international and local studios, the city is turning into a "dystopia gone right" and is now the most interesting place in the USA for new architecture.
Rohan Silva, former senior policy advisor to UK prime minister David Cameron and founder of creative workspace company Second Home, is among those who believe LA is the new destination for the US design scene.
"I think that there are three reasons why LA is really working right now," Silva told Dezeen in an interview. "One is property prices. You can have space there. If you're an artist, you can have a big artist studio and a place to build."
Images courtesy of City of West Hollywood.