London Ambulance Service took at least 20 people to six hospitals after treating a number of people at the London Bridge.
BBC reporter Holly Jones was at the bridge when the van came through and reported the van veered off the road and into crowds of people.
Armed police placed London Bridge on lockdown after the white van careered into crowds, while officers raced to the scene of the stabbings outside a pub in Borough Market.
British police said on social media that both incidents have been declared as acts of terrorism.
The BBC reports police are searching for three suspects who could be armed, according to Reuters.
Prime Minister Theresa May is returning to Downing Street to be briefed by security officials and is understood to be preparing to chair a meeting of the Government's emergency response committee - Cobra - later today.
On Twitter, Trump sent out several messages, promising to do "whatever the United States can do to help out".
Heaven also said he heard reports of stabbings.
"Now the area is completely closed with police cars going one way and ambulances going the other", he told Reuters.
He told the station: "I didn't see the van mount the kerb but I saw everything else".
The UK is still reeling from a suicide bombing in Manchester earlier this month, which killed 22 people, many of them children, following a concert.
A taxi driver named Chris told LBC that he saw three men get out of the van after it had rammed the pedestrians on London Bridge, Sky News reported.
Several witnesses also reported hearing gunshots around London Bridge. He hit about two people in front of me and then three behind. Members of the public were told by police to "run as fast as they could" westbound.
He added: "I've been in situations like this before, I don't think you realise how scary how possibly be until you put yourself into it, a police man could have shot me because it was me, the police and these three men so I was lucky definitely".
"He swerved right 'round me and then hit about five or six people".
"After that I heard what sounded like gunfire from Borough Market".
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.