A number of officers can be seen pointing their guns towards two men grappling on the ground before one of the officers grabs the man on top by his jumper and pulls him free.
The remaining man, who appears to be the suspect, can be seen trying to get up before he appears to be hit by gunfire.
A separate video, recorded from a different angle, shows another man, dressed in a suit and tie, slowly walking away with a knife in his hand.
Metropolitan Police said several people had been injured and a man had been detained following a stabbing on the bridge.
Officers were called to reports of a stabbing at a premises near London Bridge at 1.58pm.
Police said the incident is currently being treated as terror-related “as a precaution”.
The force confirmed one man had been shot by police.
London Bridge station and nearby shops and businesses have been evacuated and a cordon is in place at Borough Market.
Other footage showed police officers aiming their guns at a white lorry sitting across the central reservation on the bridge.
Several officers surrounded the vehicle before moving to the rear to check its container.
London Ambulance Service declared a major incident, saying in a statement: “We have now declared a major incident and have a number of crews at the scene in London Bridge. We will share more information as soon as we have it.”
The Metropolitan Police said: “Police were called at 1.58pm to a stabbing at premises near to London Bridge. Emergency services attended, including officers from the Met. A man has been detained by police. We believe a number of people have been injured. Further info to follow.
“We are aware of reports circulating on social media. We will release facts when we can – our info must be accurate.”
The Independent understands the man seen on the ground in footage is the person detained by the police.
Boris Johnson, the prime minister, said: “I’m being kept updated on the incident at London Bridge and want to thank the police and all emergency services for their immediate response.”
The home secretary, Priti Patel, added she was “very concerned”.
London Bridge was one of the scenes of a terror attack on 3 June 2017 when three Islamist extremists killed eight people by driving a van onto the pavement of the bridge before stabbing people in Borough Market.
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