"From 2208 hrs (local time) officers responded to reports of a vehicle in collision with pedestrians on #London Bridge".
British Transport Police has asked people to avoid the area as it deals with the situation.
The white van plowed into pedestrians on or near the bridge.
Another man who was nearby told Sky he witnessed emergency vehicles rushing to the scene.
"After that (came) the whole shooting match of ambulances and police".
"Now the area is completely closed with police cars going one way and ambulances going the other", he told Reuters.
The possible attacker was shown on the ground outside the Wheatsheaf pub in Borough Market near London Bridge.
One terrified woman said she was at a restaurant and police swarmed on the area.
Police sent out security advice to Londoners on Twitter saying "run, hide, tell" if they were caught in an attack.
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 USA court system.
Witnesses speaking to Fox News described people being ordered by police to evacuate apartment buildings and other structures, apparently to aid in their search for suspects. A BBC reporter on the bridge at the time said she saw police arrest a shirtless man just after the incident.
"He swerved right around me", Jones said, "and then hit about five or six people".
The man with the knife stabbed a waitress, who was hiding behind a partition, in the neck and stabbed a man in the back before running out of the restaurant, the witness said.
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.
Police also responded to an incident in Vauxhall in south London but later said it was not connected to the first two.
"We can confirm we have taken at least 20 patients to six hospitals across London following the incident at London Bridge". Armed police came in. The police rushed to the area but no further details were giving.
A taxi driver called Chris told LBC said he saw men armed with foot-long knives after a van drove on to the pavement.