The shocking incident happened last Wednesday at the junction of Cromwell and Warwick roads in west London.
Grandmother Helen Holland, 81, was rushed to hospital and is still fighting for her life after the accident.
The Independent Office for Police Conduct (IOPC), which investigates complaints against officers in England and Wales, has opened an inquiry.
“The woman, who is in her 80s, sustained serious injuries and was taken to hospital, where she remains in a critical but stable condition,” said the IOPC on Friday.
Its regional director Amanda Rowe added: "While some witnesses to the collision have been identified, it’s important that as part of our inquiries we speak with anyone who witnessed or recorded any part of this incident to help us establish the full circumstances."
Witnesses said Helen was carrying shopping bags as she approached the traffic lights at the “dangerous” junction at about 3.20pm.
She stepped into the road to see around an electrical box when she was hit by a police officer on a motorcycle, they added.
The force of the collision threw her across the road. There were no other casualties.
Buckingham Palace said the Duchess’s “thoughts and prayers” were with Helen and her family. They said the Duchess was grateful for the “swift response” of emergency services and she promised to “keep abreast” of developments.
The police have also made a referral to the Directorate of Professional Standards, which investigates officers’ conduct.
The Duchess, a full-time working member of the royal family, joined in events over the coronation weekend including the concert, where she was seen dancing to Lionel Richie.