It is understood that the firms are pitching for the job are Kier's refurbishment arm Wallis, Gardiner & Theobald Management Services and Schal. The cost of the scheme has not been announced.
Palace sources said three firms had been interviewed at Buckingham Palace by the Royal Household, which oversees royal construction work.
A palace spokesperson confirmed that work was about to start but refused to comment on the names of bidders.
The project team includes restoration architect Martin Ashley. Others understood to be working on the refurbishment include QS Huntley Cartwright and structural engineer Gifford & Partners.
The job is likely to involve a complete overhaul of the residence, which does not have central heating. Some ground-floor rooms are expected to be left as they were during the Queen Mother's lifetime and opened to the public during summer months.