A car has reportedly mounted the pavement on London Bridge before a stabbing attack, according to witnesses.
Witnesses said a white van veered off the road before hitting pedestrians.
Police have confirmed they are dealing with an incident and asked people to avoid the area.
Armed police have been seen at the scene and the bridge is closed in both directions.
Will Heaven was at the scene shortly afterwards and wrote on Twitter that he had seen casualties - one on the pavement and one on the edge of the road.
Reuters reported a witness saying she had seen people with "apparent knife wounds" on London Bridge.
A witness told Sky that he had seen a man lying on the ground who "looked in a bad way" and managed to flag down one of the police vans "whizzing past".
"Officers jumped out and started tending to the guy on the floor and about three of them also started running towards the London Bridge area."
London Ambulance have said: "We're responding to this incident and more information will follow when we have it."
Sky News Home Affairs Editor Mark White said there were reports of multiple injuries and some of these are serious.
He added: "Certainly reports we're getting suggest that this may be terror related but that is not being confirmed by police at this stage.
"However it is being declared a major incident.
"London Bridge would have been very busy with pedestrians at this time of night."
London Transport say London Bridge station is also closed.