The firm is up against Kier to take on a £28m construction contract and is understood to be favourite for the job, according to insiders.
The redevelopment of the hospital complex will see extensions to east and west sides of the existing main building. Works also include a new entrance and a glass roofed reception lounge linking all outpatient areas.
There will also be a relaxation area with class leading catering facilities, an expansion of the physiotherapy area and considerable refurbishment throughout.
Hospital Director Jane Abbott said that the scheme would see Benenden become a first-class independent hospital.
Some of the work will be funded from land gain for 24 homes at the site.
Benenden Hospital is an independent trust hospital with charitable status. The hospital was originally built for the National Association for the Establishment and Maintenance of Sanatoria for workers suffering from Tuberculosis.