The driver accused of mowing down crowds on Flinders Street attributed his alleged actions to "the treatment of Muslims," according to police.
A four-year-old boy with head injuries and an 83-year-old man are among 19 people injured in the incident at Flinders Street on Thursday afternoon when a 4WD allegedly drove into pedestrians.
The driver, a 32-year-old Australian citizen of Afghan descent, was arrested after being dragged out of the car by an off-duty police sergeant who is having surgery for a hand injury on Friday.
Asked if there were links to terrorism, Acting Police Chief Commissioner Shane Patton said: "That's certainly one area we're exploring in respect to motivation."
Mr Patton did say the man was not known to intelligence agencies.
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The man will undergo a psychiatric assessment and police hope to formally interview him on Friday afternoon.
Mr Patton said the man was on a mental health plan and was due to attend an appointment on Thursday morning but didn't show.
More witnesses came forward on Friday, telling how the car raced at up to 100 km/h through pedestrians lined up at a busy crossing which is used by up to 600,000 people during the morning and afternoon peak hour.
The act, deemed "deliberate" by police, comes 11 months after six people were killed in a similar incident just four blocks away in Bourke Street.
"The only thing that slowed him down was him hitting pedestrians," witness Jim told ABC TV of Thursday's incident.
Prime Minister Malcolm Turnbull held a press conference on Friday and said the attack was a "despicable and cowardly act."
"I want to emphasise because investigations are continuing, apart from that statement, there are no known links to any political issues or any, certainly any links to extremist groups," he said.
"The position I am advised of at the moment is that no terrorism link has been identified at this stage."
Premier Daniel Andrews condemned the incident as "an act of cowardice, perpetrated against innocent bystanders" but urged the public to continue to go about their festive activities.
Mr Patton reassured the public hundreds more police would be on the streets over Christmas and New Year.
All roads in Melbourne's CBD reopened about 2am and trams were operating as normal on Friday morning.