Det Con Corbett was knocked down as a suspect drove off
Det Con Catherine Corbett, 39, of Solihull, suffered head injuries in the incident in Hayes, west London.
The officer, who has been in the West Midlands force for 18 years, was trying to arrest the man when he escaped by car and knocked her over at 0800 GMT.
The Metropolitan Police is treating the incident as attempted murder.
Det Con Corbett underwent surgery at Charing Cross Hospital on Friday afternoon and her family are with her at her bedside, West Midlands Police said. The hospital is a specialist for neurosurgery.
Car abandoned
The suspect is described as being a white man aged in his 20s.
Meanwhile, police said a woman had presented herself to officers during the day and was being questioned about the incident.
The suspect was being sought as part of a fraud investigation
Det Con Corbett, who was in plain clothes, was attempting to arrest the man outside the Sheraton Hotel, near Heathrow Airport.
The suspect crashed the car into a black Ford Focus, damaging its front bumper and bonnet, after the officer was struck.
Ervis Topalli, who works at a nearby car-wash, said one of the car's tyres was punctured in the collision.
He said: "I didn't think anything of it at the time. I thought it was just a car wheel spinning.
"But then I heard the noise of the brakes and the flap of the flat tyre and I wondered what was going on."
The suspect escaped in a blue Peugeot 207, which was later found abandoned in nearby Harlington Road.
Det Con Corbett was with other officers, two of whom suffered minor injuries.
A West Midlands police spokeswoman said: "[Det Con Corbett] was with a team from West Midlands Police in Hayes to effect an arrest in connection with an ongoing fraud investigation."
Det Con Corbett is a serving member of the economic crime team of the force.
DC Catherine Corbett was seriously injured in the incident.  

A 42-year-old man and a woman in her thirties have been arrested after a police officer was left critically injured after being hit by a car as she tried to make an arrest in west London this morning. 

Police are treating the incident as attempted murder.

Detective Constable Catherine Corbett of West Midlands Police was trying to arrest a man at Sheraton Hotel, near Heathrow Airport when the suspect escaped by car and left the officer with head injuries. She was in London as part of a fraud investigation. 

The man was arrested by firearms officers in Birmingham and will be taken to London for questioning. The woman has been held after she contacted the police.

Speaking about Corbett, Assistant Chief Constable Gareth Cann said, "Cathy is an experienced police officer and a very good police officer."

Police have appealed for anyone with any information to contact the Metropolitan Police Incident Room on 020 8358 0400 or Crimestoppers anonymously on 0800 555 111.