The suspect gave chase shortly after 5pm local time in response to alleged criminal mischief, police said, refusing to rule out terrorism as a possibility.
SBS Senior Correspondent Brett Mason was on the scene in New York City, and several eyewitnesses he spoke to said: "it wasn't terrorism: just a lunatic with a cleaver."
Department chief James O'Neill, who is poised to take over as New York police chief, said: "At this point, the initial radio run was (for) criminal mischief."
"As part of our investigation, nothing is off the table," he added.
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Akram Joudeh, 32, tried to remove a parking boot from a car not far from Penn Station, then fled from officers on foot after pulling 28cm meat cleaver from his waistband, police said.
An initial attempt to taser him failed.
When he mounted the front grille of a marked police car, a detective in civilian clothes attempted to subdue him.
The off-duty officer sustained a six-inch gash from his temple to his jaw, at which point police opened fire 18 times.
The suspect is in critical but stable condition in the hospital, the detective is in serious condition and two more officers sustained non-life-threatening injuries, O'Neill said.
Asked by reporters whether it was necessary to fire 18 shots in a busy area, O'Neill replied: "Keep in mind that he had just attacked an off-duty officer who has a 6-inch gash on his face and he has a 11-inch cleaver."
With AFP.
