A white van was said to be traveling at about 50 miles per hour when it veered off the road and on to the sidewalk on the bridge late Saturday night and struck pedestrians, less than a month after the Manchester Arena was bombed.
He said the pub was then closed and armed police told them to leave the area.
The agency additionally tweeted a third incident has been reported in the Vauxhall neighborhood of South London, some 20 minutes away by vehicle. The BBC cited police as saying there had been more than one fatality.
Transport for London said busy London Bridge station was closed at the request of police.
United Kingdom police are responding to incidents in London Bridge and Borough Market in the UK.
European security official says it is too soon to say whether incidents around London Bridge are militant attacks.
A third incident is taking place at Vauxhall, Scotland Yard said.
Prime Minister Theresa May has been briefed on the incident and will hold an emergency meeting with members of the government on Sunday.
London Ambulance have said: "We're responding to this incident and more information will follow when we have it".
The city's transportation authority, Transport for London, shut down the bridge in both directions.More news: Serena's back at a Slam! OK, sort of; she watched Venus win
Several witnesses also reported hearing gunshots.
"It looked like he was aiming for groups of people". US President Donald Trump has also been informed, his press secretary announced.
The UK is still reeling from a suicide bombing in Manchester earlier this month, which killed 22 people, many of them children, following a concert.
Witnesses on London Bridge reported seeing a van mounting the pavement and hitting pedestrians and a man with a knife running.
"I saw people moving, but we though, f--, this is a serious thing", Al-Ostta said.
Taxi driver Chris was on London Bridge as the incident unfolded.
"He swerved right around me", Jones said, "and then hit about five or six people". They did not say if officers fired the shots.
Martin Lucas, who posted images from the site of the attack on social media, said police had quickly locked down the area. "And then people started screaming coming back up", the 31-year-old said. Witness Elsbeth Smedley told CNN that a man ran into Elliot restaurant in Borough Market and stabbed two people before running out.
Indians are also speculated to be in the area as there is an India vs Pakistan match scheduled at the Edgbaston Cricket Stadium in Birmingham on June 4.
"Then three men got out with long blades, 12 inches long and went randomly along Borough High Street stabbing people at random".
A witness on the bridge told CNN the van mowed down pedestrians as it sped south across the bridge, leaving bodies lying in the roadway.
In March, four people were killed in London after Khalid Masood rammed his vehicle into pedestrians on Westminster Bridge, before fatally stabbing a policeman outside the Houses of Parliament.