Several people have been injured with one feared dead as gunfire was also heard in the area, police said.
"From 2208 hrs (local time) officers responded to reports of a vehicle in collision with pedestrians on #London Bridge. Officers have then responded to reports of stabbings in #BoroughMarket. Armed officers responded and shots have been fired," Metropolitan Police said in a statement on Twitter.
"Officers are now responding to an incident in the #Vauxhall area," the statement added.
"The Prime Minister is in contact with officials and is being regularly updated on the incident at London Bridge," Downing Street said.
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Transport for London said the bridge has been closed in both directions due to a "major police incident".
London Ambulance Service tweeted: "Multiple resources attending an incident at #LondonBridge, please avoid the area".
Holly Jones, a BBC reporter who was present at London Bridge at the time of incident said a van veered off the road into a crowd of pedestrians.
"A white van driver came speeding - probably about 50mph - veered off the road into the crowds of people who were walking along the pavement," she told BBC News.
A large number of police officers were seen gearing up with helmets and riot gear, believed to be on the hunt for suspects. There were also reports of some people being taken into police custody.