Christopher Dorner was shot at by a fish and game ranger, as he fled in a stolen vehicle in north-east Los Angeles, Reuters is reporting.
The agency states that earlier Dorner had broken into a home, tied up a couple and stole their pick-up truck.
Local media is also reporting it is believed one of the hostages was able to flee and raise the alarm.
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Investigators believe Mr Kushner has significant information relevant to their inquiry, officials told NBC News.
The men were detained as part of an investigation by MI5 and the Metropolitan Police's Counter Terrorism Command.
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