London Ambulance Service tweeted: "Multiple resources attending an incident at #LondonBridge, please avoid the area".
BBC reporter Holly Jones was at the bridge when the van came through and reported the van veered off the road and into crowds of people.
"Incidents at London Bridge and Borough Market were declared as terrorist incidents", tweeted United Kingdom police.
Armed police are there, as well as at nearby Borough Market, where another incident has been reported, according to a Twitter post by the Metropolitan Police.
There are fears of another terror attack in the British capital after a van drove into crowds of people at London Bridge on Saturday night.
A stabbing at nearby Vauxhall was unrelated, the Met added.
Police boats were searching the river - apparently looking for people who may have been thrown off the bridge, she said.
Prime Minister Theresa May is returning to Downing Street to be briefed by security officials and is understood to be preparing to chair a meeting of the Government's emergency response committee - Cobra - later today.
- London Bridge was closed in both directions.
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An eyewitness also described to CNN hearing what "sounded like" gunshots. In the U.S., President Donald Trump argued in favor of his controversial travel ban as London authorities responded to the reports.
Some netizens witnessed the incident tweeted that they saw people got injured and the police has come to deal with it.
"The sheer sense and waves of and kind of panic that run through people. the police, I can't stress how well they've handled it but it takes them a few minutes to actually get to places", Mr Dastyari said.
Police rushed to London Bridge late Saturday after a vehicle mowed down pedestrians on the bridge.
Witness Nick Archer told Sky News 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". However, it was not clear if the shots were fired by police or an assailant or assailants.
The Sun newspaper reported a group of five men wearing stab-proof vests began "attacking revelers with hunting knives" after leaving the van.
Pence also tweeted, "Our thoughts & prayers are w/ the victims, courageous first responders & all the people of London".
The patrons sheltered in the basement and police have arrived at the scene, the witness said. He says he subsequently witnessed three attackers stab at least four people, including a woman in her early 20s.
After ramming the auto into a barrier outside the Houses of Parliament, the driver, 52-year-old Khalid Masood, got out and stabbed a police officer to death before being gunned down by responding officers.