Counter-Terrorism Police North West said three men – aged 29, 26, and 21 – had been detained in the Kensington area of the city and arrested under the Terrorism Act in connection with the incident.
They said they were continuing to keep an open mind about the cause of the explosion and are working closely with Merseyside Police, who were called to the blast outside Liverpool Women's Hospital at 10.59am.
The male passenger of the car was declared dead at the scene and is yet to be formally identified. The driver, also a man, was injured at the scene and remains in hospital in a stable condition.
"So far we understand that the car involved was a taxi which pulled up at the hospital shortly before the explosion occurred," a spokesman for Merseyside Police said.
"Work is still going on to establish what has happened and could take some time before we are in a position to confirm anything.
"We are keeping an open mind as to what caused the explosion, but given how it has happened, out of caution, counter-terrorism police are leading the investigation, supported by Merseyside Police.
"Our response is ongoing at the hospital and will be for some time. Cordons are in place and there are some road closures."
Police said a major incident had been declared.
Images of a vehicle on fire, and later burnt out, at the scene were earlier shared online.
Footage of explosions and billowing smoke outside the hospital were also shared as a bomb disposal unit attended.
Liverpool Women's Hospital said visiting access had been restricted "until further notice" and patients had been diverted to other hospitals "where possible".
"We are reviewing our patient activity for the next 24 to 48 hours and patients should wait to be contacted for updates about any planned appointments or other attendance at the hospital," it said.
"Our staff are being permitted to leave and enter the hospital under the supervision of Merseyside Police.
"We would like to say thank you for the co-operation and support of our patients, visitors and staff who have been impacted by this incident, in particular those members of staff who were at the scene at the time of the incident."
Phil Garrigan, chief fire officer of Merseyside Fire and Rescue Service, said the car fire at Liverpool Women's Hospital was "fully developed" when two engines arrived shortly after 11am.
"The operational crews extinguished the fire rapidly but as has been reiterated by the police chief constable, there was one fatality," he said at the scene.
"Another individual had left the vehicle prior to the fire developing to the extent that it did. Our thoughts are with them and the families of those involved."