The man, 38, was arrested and taken to Albury hospital under police guard after the incident, which occurred in Wodonga Place about 7.30am.

The man remained under guard on Friday night.

A female passenger had to be freed after being trapped in the black BMW with serious leg injuries.

She was treated at the scene, before also being taken to hospital with suspected fractured legs.

The bus driver and passengers were not injured.

Detective Acting Inspector David Forland said the car had been travelling north on Wodonga Place when it failed to negotiate the Smollett Street roundabout and crashed into the rear of a stationary bus.

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The driver left the site but was chased by witnesses, who held him until police arrived.

"It wasn't a very long run from the scene, they chased him up to the old ambulance station there and they managed to detain him there," Detective Acting Inspector Forland said.

"I commend them for their actions, to actually get involved in an chase.

"We don't expect that from members of the public however it was greatly appreciated in these circumstances."

No witness was injured.

The driver was arrested and taken to hospital for mandatory blood and urine testing as well as treatment for injuries.

Police are investigating whether the car was stolen and if speed was a factor in the crash.

Witnesses suggested the car had driven at speed over Lincoln Causeway.

Nearby resident Tony Love-Linay described waking to a large bang and screams about 7.30am.

An Ambulance NSW spokeswoman said the woman, aged in her 40s, was taken to hospital in a stable but serious condition with leg and pelvic injuries and would be assessed for compound fractures.

The spokeswoman said five ambulance crews had been deployed at first.

Detective Acting Inspector Forland agreed the incident could have been worse in a busy central location.

"With that area at that time of the day, there's pedestrians, there's a lot of traffic flow through there," he said.

"Certainly there was a high potential for further injuries there."

The bus and car had to be towed from the scene and Wodonga Place between Smollett and Dean Streets remained closed for several hours.

Fire and Rescue NSW and Albury and Border Rescue Squad also assisted.

Anyone who saw the car before the crash is asked to contact Albury police on (02) 6023 9299.