The unnamed woman was found unconscious at the King Edward VII Hospital, where the Duchess spent three nights earlier this week being treated for severe pregnancy sickness.
Paramedics tried to revive her but she was pronounced dead at the scene.
Police said the cause of her death was "unexplained" but they were not treating it as suspicious.
According to the Daily Mail, the circumstances in which she was found indicated she might have killed herself.
A spokesman for the Metropolitan Police said: "Police were called at approximately 9.25am on Friday, December 7, to a report of a woman found unconscious an address in Weymouth Street, W1.
"London Ambulance Service attended and the woman was pronounced dead at the scene.
"Inquiries are continuing to establish the circumstances of the incident.
"The death is not being treated as suspicious at this stage."
A spokesman for the hospital said a statement was expected shortly.
The dead woman is reported to be a nurse who was working on reception who took the call at 5.30 on Tuesday morning from two presenters from the Sydney radio station 2Day FM, who pretended to be the Queen and the Prince of Wales.
The unsuspecting hospital worker put the call straight through to the Duchess's ward, where a nurse spent two minutes chatting about the Duchess's condition and treatment.
The hospital described the incident earlier this week as "deplorable"