Officers carried out a controlled explosion on the car which was loaded with petrol, gas cylinders and nails at 0200 BST in Haymarket.
Police were alerted by an ambulance crew who spotted smoke coming from the silver Mercedes, which was parked near the Tiger Tiger nightclub.
"International elements" are believed to be involved, the BBC has been told.
Deputy Assistant Commissioner Peter Clarke, head of Scotland Yard's counter-terrorism command, said: "It is obvious that if the device had detonated there could have been significant injury or loss of life."
The area was cordoned off while police examined what they described as a "potentially viable explosive device".
Prime Minister Gordon Brown said Britain faces "a serious and continuous threat".
He added the public "need to be alert" at all times.
The prime minister's comments were echoed by the new Home Secretary Jacqui Smith who has chaired a meeting of the government's emergency unit Cobra.
The BBC's Andy Tighe said the timing was significant coming a day after Gordon Brown became prime minister, and with the second anniversary of the 7 July bombings approaching.
A witness reported seeing gas canisters being removed from the car, a silver Mercedes, at around 0400 BST (0300 GMT).
Bouncers from a nearby nightclub said they saw the car being driven erratically before it crashed into a bin. They said the driver then got out and ran off.
Police sources have confirmed that gas canisters were involved in the incident, close to Piccadilly Circus.
The police source said the bomb was a "big device" and posed a real and substantial threat to the area around Haymarket, which is in London's theatreland.
Dozens of forensic officers examined the scene and the car was removed for further examination.
Scotland Yard said detectives from Counter Terrorism Command were investigating the potential bomb plot and will be checking the CCTV in the area.
Police are believed to have also carried out a search of other key areas in the capital shortly after the discovery of the car.
A spokesman said: "Police were called to reports of a suspicious vehicle parked in Haymarket, shortly before 2am this morning.
"As a precautionary measure the immediate area was cordoned off while the vehicle was examined by explosives officers.
"They discovered what appeared to be a potentially viable explosive device. This was made safe."
Police say Haymarket is likely to remain closed for some time and severe travel disruption is predicted.
Piccadilly Circus Tube station has reopened after being closed for some time.
The BBC's Daniela Relph, at the scene, said the heart of London was completely closed off and police officers were concentrating on keeping people away.
'No intelligence'
Professor Paul Wilkinson, a terrorism expert, said a passer-by had tipped off the police and officers would be concerned they did not have prior intelligence.
The current terrorism threat level has been classed as severe - meaning an attack is highly likely - since 14 August 2006.
Intelligence sources said they were keeping an open mind on who was responsible for the car bomb.
The BBC's security correspondent Frank Gardner said the incident had "come from nowhere" and that the driver of the car was now Britain's "most wanted".
He added that the police would be in charge of the investigation and that CCTV was the key to finding the "first clue" as to who was behind the attempted bombing.
John O'Connor, former commander of Scotland Yard's Flying Squad, told BBC News the incident bore all the "hallmarks" of a failed suicide bomb attempt. And it was "lucky" the police had received the tip-off .