One person has been detained by Scotland Yard following the incident in Exhibition Road in South Kensington.
"It is believed that a number of pedestrians have been injured," Metropolitan Police said in a statement.
Officers are on scene, and the London Ambulance Service have been called, it said.
"A man has been detained at the scene. Enquiries to establish the circumstances are underway," the statement said.
It is not yet clear if what is being described as an "incident" by police is terrorism related.
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A Downing Street spokesperson said British Prime Minister Theresa May is "being kept up to date on events".
A BBC reporter was leaving the Natural History Museum as the incident occurred.
"I could see a car diagonally across the road, looking like it was going into one of the boulders on the side of the road, and I could see a crowd of people around what was clearly one or two people on the pavement," she said, confirmingthere were armed police at the scene.
Reports suggest theblack Toyota Prius which mounted the pavement was a licensed taxi cab.
Today's incident follows a number of attacks in which vehicles have been used to target pedestrians in London, including at London Bridge in June which claimed eight lives.
In March, a terrorist rammed his vehicle into the side of the Parliament building in the British capital killing four people.