Stagecoach has said its bus moved away from the scene after the 60-year-old woman was injured on Prince Edward Road in South Shields on Wednesday morning.
She was taken to Newcastle's Royal Victoria Infirmary hospital for treatment.
Northumbria Police has said it is investigating the collision, which happened at 7.20am, and that the driver of the bus will be spoken to by police as information is piece together about the collision.
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It has said the woman is due to be discharged from hospital in coming days.
A spokesman added: "We would still like to hear from anyone who witnessed what happened as inquiries continue."
A spokesperson for Stagecoach North East said: "We can confirm that around 7.25am on Wednesday, February 7, a pedestrian was involved in an incident with one of our vehicles on Prince Edward Road in South Shields.
"At the time, the driver was unaware of the incident as the pedestrian was stood in the central reservation and appears to have fallen into the side of the bus as it was passing, and therefore did not stop immediately.
"A member of the Stagecoach management team assisted at the scene and a review of CCTV footage is under way."
The North East Ambulance Service said it sent a double-crewed ambulance, a hazardous area response team and a rapid response vehicle to the scene.
Anyone with information which could help Northumbria Police is asked to call 101 or Crimestoppers anonymously on 0800 555 111.