Witnesses said people had been trapped underneath the single-decker Arriva bus before emergency services arrived.
County Durham and Darlington fire and rescue service said three people had been injured in the crash.
A spokeswoman for the North East ambulance service said paramedics were at the scene. “We have sent the air ambulance, three double-crewed ambulances, hazardous area rapid response team and an officer,” she said. “We have reports of three injures, one seriously. Two patients have left for hospital.”
Police, fire and ambulance services closed the road from the A167 Northgate junction to the Crown Street junction while they dealt with the accident.
An eyewitness told the Northern Echo that they believed that one person had died. “The number 2 bus was parked up on the side of the road and pulled out before going straight into the bank,” the witness said. “There was someone stuck under the wheel – I think two people were trapped, they’d just been walking along the road.
“I could see the person had passed away, they were not moving, and after a while, they were covered up with a green blanket, the head was under the wheel and their legs and arms were just spread out.
“It was absolutely shocking, I’m still shaking now – it just pulled out. That person was just out doing a bit of shopping and now they’re dead.
“I don’t know if anyone in the bank was hurt, most of them were out and were taken into Boots. Everyone on the bus is giving statements. It was so horrible.
“The driver was sat close by and has been talking to the police. My son and I were so lucky, we were using the cash machine just seconds before, but I’d forgotten my card, so left. If I hadn’t left it at home, I’d still have been there when it happened.”