Armed police rushed to the scene and shot dead the three male attackers in the Borough Market area near the bridge, as authorities urged Londoners on Twitter to "run, hide, tell" if they were caught up in the violence.
The attacks came just five days ahead of a national parliamentary election on Thursday. The ruling Conservative Party suspended national campaigning on Sunday, a spokesman for Prime Minister Theresa May said.
"Following updates from police and security officials, I can confirm that the terrible incident in London is being treated as a potential act of terrorism," May said in a statement as events unfolded.
The killings also occurred less than two weeks after a suicide bomber killed 22 people at a pop concert by U.S. singer Ariana Grande in Manchester in northern England.
There was no immediate claim of responsibility.
"Sadly, six people have died in addition to the three attackers shot by police," Mark Rowley, Britain's top anti-terrorism officer, said. The three attackers had been wearing what looked like explosive vests that were later found to have been hoaxes.
The BBC showed a photograph of two possible attackers shot by police, one of whom had canisters strapped to his body. Hours after the attack the area remained sealed off and patrolled by armed police and counter-terrorism officers.
The London Ambulance Service said 48 people had been taken to five hospitals across the city and a number of others had been treated at the scene for minor injuries.
Streets around London Bridge and Borough Market, an area that is a major transport hub and is packed with bars and restaurants, was busy with people on a night out.
London Mayor Sadiq Khan spoke of a "cowardly attack on innocent Londoners and visitors to our city enjoying their Saturday night".
Witnesses described a white van careering into pedestrians on the bridge.
"It looked like he was aiming for groups of people. I froze because I didn't know what to do," Mark Roberts, 53, a management consultant, told Reuters. He saw at least six people on the ground after the van veered on and off the pavement.
"It was horrendous," he said.
A taxi driver told the BBC that three men got out of the van with long knives and "went randomly along Borough High Street stabbing people."
Witnesses described people running into a bar to seek shelter.
"People started running and screaming, and the van crashes into the railing behind. We went toward Borough Market and everyone went inside (the bar). Everyone in the bar started pushing people from the exits," one witness, who gave his name as Brian, 32, told Reuters.
Another witness, who declined to be named, his white top covered in blood, described a scene of panic in the bar.
"They hit the emergency alarm. There was a line of people going down to the emergency exit. And then people started screaming coming back up," the 31-year-old said.
"Around the corner there was a guy with a stab wound on his neck ... There was a doctor in the pub and she helped him. They put pressure on the stab wound."
BBC radio said witnesses saw people throwing tables and chairs at the attackers to protect themselves.
The BBC showed dozens of people being escorted to safety through a police cordon with their hands on their heads.
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Islamic State, losing territory in Syria and Iraq to an advance backed by a U.S-led coalition, sent out a call on instant messaging service Telegram early on Saturday urging its followers to launch attacks with trucks, knives and guns against "Crusaders" during the Muslim holy month of Ramadan.
Similar attacks, in Berlin, Nice, Brussels and Paris, have been carried out by Islamist militants over the past two years.
May was due to chair a meeting of the Cobra security committee later on Sunday.
U.S. President Donald Trump took to Twitter to offer U.S. help to Britain. The White House said he had been briefed on the incidents by his national security team.
French president Emmanuel Macron said on Twitter that "France is standing more than ever side by side with the UK".
The incident bore similarities to a March attack on Westminster Bridge, west of London Bridge, in which a man killed five people after driving into a crowd of pedestrians before stabbing a police officer in the grounds of parliament.
Several witnesses also reported hearing gunshots around London Bridge.
"We were in an Uber (taxi) going toward London Bridge and suddenly we saw people running. The Uber stopped, we asked people what was going on – people said there was shooting," said Yoann Belmere, 40, a French banker living in London.
"Now the area is completely closed with police cars going one way and ambulances going the other," he told Reuters.
The Manchester bombing on May 22 was the deadliest attack in Britain since July 2005, when four British Muslim suicide bombers killed 52 people in coordinated attacks on London's transport network.