Officers were called at approximately 11:00am on Friday, 16 April following reports of a collision involving a pedestrian and moped on Maple Road, SE20.
The pedestrian, aged in his 70s, was taken to a south London hospital where he remains in a critical condition. His family have been informed.
The 24-year-old rider of the moped stopped at the scene. He was arrested and taken to a south London police station where he remains in custody.
Detectives from the Roads and Transport Policing Command are investigating. They would like to speak with anyone who witnessed the collision or who captured it on dash cam.
Anyone with information is asked to call the Serious Collision Investigation Unit based at Catford on 020 8285 1574.