Trump expressed support for both the city of London and the broader United Kingdom while reiterating his call for a travel ban that has been consistently struck down by the US court system.
A witness told the BBC that the van was driving at high speed on the wrong side of the road.
One woman told Reuters she saw what appeared to be three people with knife wounds and possibly their throats cut at London Bridge at the Thames river.
Police have so far been scant with details, saying they will release information when they can. The condition of the stabbed people was not known.
Some passersby told The Spectator's Will Heaven they saw 15-20 people struck, while others said they saw 7-8.
It swerved into oncoming lanes before hitting a bus stop and coming to a stop, Roberts said.
"It looked like he was aiming for groups of people".
"Then we heard three gunshots, definitely gunshots, and we ran". He estimated 100 people were on the bridge at the time.
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The cafe stabbing occurred in Borough Market, a popular market with gourmet food and produce stalls and several restaurants inside.
He said trains were not stopping at London Bridge and Borough Market.
"Then we hid under the table and people came into the restaurant and knocked a bunch of stuff over, like the till". Another man was seen lying on the ground a short distance away in the photograph. She witnessed the injured being treated by paramedics, she said. Police apparently told other pedestrians in the area to flee. "One of the officers jumped out and started attending to the guy on the floor".
"The United States stands ready to provide any assistance authorities in the United Kingdom may request", State Department spokeswoman Heather Nauert said. They asked Londoners to "remain calm but be alert and vigilant". We need the courts to give us back our rights.
The US embassy in London tweeted: "Please avoid the area and monitor local news/@metpoliceuk for updates". London Transport Police said that the London Bridge station is closed off.
Grande visited victims of the bombing in the hospital Saturday and is scheduled to perform at a star-studded benefit concert in Manchester on Sunday.
BBC reports more than one person has been killed.
In March, a British convert to Islam ran down people with a vehicle on Westminster Bridge, killing four, then stabbed a policeman to death outside Parliament.