Hawking, 62, is under surveillance in a London hospital's intensive care unit with pneumonia, but his family is "worried sick" he has again been the victim of a series of assaults, the newspaper said.
Police opened their second investigation in four years into the series of mishaps he has sustained.
The millionaire Hawking, author of the best selling, "A Brief History of Time," is paralyzed with a neurological disease, needs round the clock care and speaks with a computerized voice box.
The latest inquiry was triggered when nursing staff contacted police last summer after he was left stranded in his wheelchair in the garden of his Cambridge home on the hottest day of the year. He suffered severe heatstroke and sunburn.
Most of the injuries he suffered were bumps and bruises, which did not require hospital care.
Detectives discovered Hawking had regularly visited the hospital's emergency department with unexplained injuries including a broken wrist, gashes to the face and a cut lip.