“Officers have then responded to reports of stabbings in Borough Market,” at the south end of the bridge.
“Armed officers responded and shots have been fired,” police said.
A third incident, in nearby Vauxhall area proved to be unrelated.
The attack came just more than a week after a suicide bomber killed 22 people at a concert in the northern city of Manchester and just days ahead of the June 8 election, where security is a major theme.
Prime Minister Theresa May said she could “confirm that the terrible incident in London is being treated as a potential act of terrorism.”
She will hold an emergency ministerial meeting later on Sunday and Facebook has activated its safety check function for people in London to let their loved ones know they are safe.
It was not immediately clear how many victims the attack claimed.
Witnesses on London Bridge reported seeing a van mounting the pavement and hitting pedestrians and a man with a knife running.
“There was a van that crashed into the fences on London Bridge. And then there was a man with a knife, he was running. He came down the stairs and went to the bar,” Dee, 26, who was visibly in shock and declined to give her last name, told AFP.
Gerard Kavanar, 46, said he had seen a chef with “blood on his shoulder” in the area.
The London Ambulance Service said “multiple resources” were being sent to the scene. AFP reporters saw two police helicopters over the area and several roads were shut down.
The police were urging the public to run to a place of safety, or hide if they cannot.
Will Heaven, managing editor of The Spectator magazine, said on Twitter he saw “two casualties — one on pavement, one edge of road” and reported seeing armed police on the bridge.
BBC reporter Holly Jones, who was there at the time of the incident, said she saw a van driven by a man travelling “at about 50 miles (80 kilometres) an hour”.
She said about five people were being treated for injuries after the vehicle mounted the pavement and hit them.
“There’s several police boats with torchlights searching the Thames at the moment,” she told BBC radio.
She added that she saw a man, who had his shirt off and was in handcuffs, being arrested by police.
Another witness, who gave his name as Alessandro, told BBC radio that he saw a van strike several people on London Bridge.
“I saw this van going left and right, left and right, trying to catch as many people as he could. And people just tried to get out of the way of the van.
“Then I tried to help people, wounded people.”
Of the casualties, he said: “Three of them, yes (were conscious), and one guy was not talking at all, was just, like, down.”
“There were five or six people that we tried to help, they were young people.”
His friend Giovanni said he contacted police and the ambulance services, which arrived within around two minutes.
Simon Johnson, who had been near London Bridge, told Sky News he had hid in the basement of a restaurant in the area for about an hour.
“Still in a state of shock to be honest,” Johnson said.
Husband and wife Ben and Natalie told BBC Radio 5 Live they were outside Borough Market when they witnessed the incident there.
Ben said: “We saw people running away and then I saw a man in red with a large blade, at a guess 10 inches long, stabbing a man, about three times.
“It looked like the man had been trying to intervene, but there wasn’t much he could do. He was being stabbed quite coldly and he slumped to the ground.”
Ben said the man then walked towards the Southwark Tavern. He said they saw a metal chair being thrown towards the man.
“Then we heard three gunshots, definitely gunshots, and we ran.”
Share this post with your Friends on