As they rampaged through a busy night-life hub around London Bridge, the trio wore what looked like suicide vests which turned out to be "hoaxes", said Mark Rowley, head of counter-terrorism policing.
The Met said in a tweet that officers had also been deployed to deal with an incident in Vauxhall.
The suspects were confronted and shot by the police within eight minutes of the first call, Rowley added.
Almost two weeks ago, a terrorist in Manchester bombed a concert put on by Ariana Grande, killing 22 people, several of whom were children.
It wasn't immediately clear whether Trump - who was briefed by his national security team, according to spokesman Sean Spicer - had additional, non-public information about the attacks at the time he sent his message on Twitter.
"At this stage, we believe that six people have died in addition to the three attackers shot dead by police".More news: MIT issues statement regarding research on Paris Agreement
Gunshots were later heard in the area while one social media user posted a photograph appearing to show one of the attackers lying on the ground outside a pub in the food market, with what seemed to be canisters strapped to his body.
London Ambulance Service said 48 people were injured. "Any attack like that is terrorism, doesn't matter who does it, doesn't matter if it's a British guy, doesn't matter if it's somebody from a Middle Eastern country, doesn't matter if they're American - it's still terrorism", passenger Cat Heathcote said.
"We were in an Uber (taxi) going towards London Bridge and suddenly we saw people running".
Simon Thompson told Sky News that he was just outside Borough Market when he saw crowds fleeing.
British Prime Minister Theresa May has called the London Bridge incident a "potential act of terrorism".
BBC radio said witnesses saw people throwing tables and chairs at the perpetrators of the attack to protect themselves.More news: Hundreds of Juventus fans injured in stampede in Turin
A witness who was on the bridge told the BBC she saw a speeding van veer onto the pavement, striking as many as six people.
Police said they received reports of a vehicle striking pedestrians on London Bridge around 10:08 p.m.
"He swerved right around me", Jones said, "and then hit about five or six people". More than 30 victims were treated at London hospitals and a number of others suffered less serious injuries.
People leave the area with their hands up after an incident near London Bridge in London, Britain, in the early hours of June 4, 2017.
In March, four people were killed in London after Khalid Masood rammed his auto into pedestrians on Westminster Bridge before fatally stabbing a policeman outside the Houses of Parliament.
Britain had just lowered its threat warning last week, following a bombing at the Manchester Arena on May 22 which left 22 dead and 64 others injured.More news: Cher leads tributes to Gregg Allman
- Rihanna Was The Real Star Of NBA Finals Game 1
- Nunes-led House Intelligence Committee asked for 'unmaskings' of Americans
- London terror attacks: What we know and don't know
- Champions Trophy: Fans await Kohli's surgical strike on Pak
- NY Times: Trump Not Planning to Invoke Executive Privilege for Comey Testimony
- German rock concert resumes after terrorism alert
- Raiders members share reactions, predictions on NBA Finals
- Real's remarkable Ronaldo again delivers when it matters
- Cavaliers need heart, pride, fewer mistakes in Finals
- Zidane team talk was key to victory, says Ronaldo