The witness said the van hit five to six people.
There were also reports of a third incident, a stabbing in the Vauxhall area of London.
"We can confirm we have taken at least 20 patients to six hospitals across London following the incident at London Bridge".
One woman said she saw three people with what appeared to be their throats cut at London Bridge.
"From 2208 hours officers responded to reports of a vehicle in collision with pedestrians on London Bridge".
Borough Market, where the stabbing occurred, is a popular food market in south London, just south of London Bridge, with gourmet food and produce stalls and several restaurants inside.
Authorities are also working a second incident at Borough Market, which is not far from London Bridge. "He hit about two people in front of me and then three behind".
"We are all saddened by this senseless tragedy. especially since it happened on home ground, and we want to help the people affected, where possible", she told The Straits Times.
Casualties were seen being taken away from Tito's restaurant on London Bridge Street with blood on the steps of the eatery.
Transport for London says the busy London Bridge station has been closed at the request of police.
Both incidents were confirmed as terrorism.
"Now the area is completely closed with police cars going one way and ambulances going the other", he told Reuters.
I'd say there are about four severely injured people.
Eyewitnesses reported panic in the vicinity of the attacks, close to a major transport hub and in an area packed with restaurants and bars.
England has been under heightened alert since a bomb was detonated at an Ariana Grande concert in Manchester nearly two weeks ago, killing 22 and wounding more than 100 as they left the venue. Five victims were killed in the attack.