Hammond, who often drives high-performance cars as part of presenting the show, has been taken to hospital following the incident at an airfield near York.
A BBC spokeswoman said: "The incident happened quite recently and the focus of our attention is on Richard at this stage. "We will make further details known as soon as they are available."
Inspector Mike Thompson, of North Yorkshire Police, said officers were dealing with the incident at the airfield north-east of York.
He said: "At 5.45pm (on Wednesday) evening we received a report via the fire service of a male person trapped in what was described as an overturned jet car which had been driven on the airfield.
"The male occupant has received serious injuries and has been airlifted to hospital at Leeds."
He said no further details were available at this time.
Hammond was in a "critical" condition at Leeds General Infirmary, a hospital spokesman said.
The 36-year-old presenter was in a special neurological unit.
A North Yorkshire Police spokesman confirmed the accident happened on Elvington airfield, which is seven miles to the south-east of the city.
He said the presenter was filming an episode of Top Gear at the airfield at the time.
The spokesman added: "Relatives have been informed and are en route to the hospital itself."
Top Gear magazine editor Michael Harvey told BBC News 24: "I know every single precaution will have been taken but something clearly absolutely unaccounted for has gone wrong, and Richard has unfortunately suffered the consequences."
James May, who co-presents the popular motor show with Hammond and Jeremy Clarkson, said he was devastated to learn his "old mate" had been injured.
Former Top Gear presenter Quentin Willson said one of Hammond's best qualities was that he has "no fear". He told BBC News 24: "It does sound very serious. It is a huge, huge tragedy. He is a wonderful, unique and distinctive Top Gear presenter."
Top Gear airs in Australia on SBS.