There were also reports of a third incident, a stabbing in the Vauxhall area of London.
He said three men got out of the van with knives and went towards the group and started stabbing them. "Our thoughts are with those who are caught up in these terrible events".
DUP leader Arlene Foster said she was thinking "of all affected by these horrific incidents in London".
The US Department of Homeland Security put out a statement saying, "At this time, we have no information to indicate a specific, credible threat in the United States".
London Bridge underground station has been closed at the request of the police, as have the Network Rail stations of Waterloo East, Charing Cross and Cannon Street.
Police said they fired shots after reports of stabbings in the nearby Borough Market area.
Videos posted on social media and witnesses told media outlets that police burst into pubs shouting at people to flee the area.
President Donald Trump and Vice President Mike Pence have been briefed about the situations in London, officials say.
Frightened people were led up Southwark Street away from Southwark Bridge Road, and police officers were seen screaming to people and ordering them to run. The proximity of the attacks and the timing seem to suggest they were coordinated, but police have not confirmed that. Frank, thanks so much for speaking with us. What are authorities telling us?
The van swerved into oncoming lanes before hitting a bus stop and coming to a stop, Roberts said. Five people were killed, including the attacker, and at least 40 people were injured.
"I saw a stream of 20 ambulances go towards London Bridge". We're talking about central London.
Police added that it was too early to suggest a motive. And that's what we know so far from London Bridge.
Jamie, a witness who was in a restaurant on Rochester Walk near London Bridge, told the Press Association: "We were in the Black and Blue restaurant, we heard a fight and everyone got up and everyone rushed out of the restaurant and we heard a massive, massive bang". K., we will be there - WE ARE WITH YOU.
Just minutes before terror struck, pubs in the popular Borough market were crammed with revellers enjoying a balmy summer night out in a trendy part of the city, many of them watching the Champions League final on big screens. Basically lots of people came running inside, we didn't really know what was going on. And that's what we have at Borough Market.
Mayor Sadiq Khan said the incident was still unfolding. Borough station is also closed and there are delays at Blackfriars. I actually was riding my bike along the Thames today upstream. At 2.13 a.m. BST (9.13 p.m. ET), London Ambulance announced that at least 20 people were taken to six hospitals in the capital. Just to situate our listeners, London Bridge would be upstream from the famous Tower Bridge, downstream from St. Paul's Cathedral.
It comes less than three months after an attacker ran down people with a vehicle on Westminster Bridge, killing four, then stabbed a policeman to death outside Parliament.
Ms Jones later reported seeing a man being arrested by police.
Another incident in Vauxhall in South London is also being simultaneously investigated - it is not suspected to be in connection to the other incidents. There has - that has to be on people's minds there. Guy's Hospital, near London Bridge, was placed on lockdown "due to the ongoing incident in central London.to keep patients, relatives and staff safe", it said in a statement.
Theresa May is on her way back to No 10. The attacker was gunned down by a police officer. That's NPR's Frank Langfitt reporting from London.