The death on Tuesday comes in the midst of an escalating federal crackdown on migrants.
The man shot dead was identified as Lorenzo Salgado Araujo, described by ICE as a Mexican national and "illegal alien" who attempted to evade arrest during a "targeted enforcement operation" by federal immigration officers.
Ronaldo Salgado, who identified himself as the son of the slain driver, told the Spanish-language television station Telemundo Houston that his father was shot while he was seeking work in the area.
According to the ICE account of the incident, Salgado "rammed an ICE law enforcement vehicle, refused to follow multiple verbal commands, and weaponised his vehicle in an attempt to run over an ICE law enforcement officer".
The confrontation resulted in "our officer firing his weapon in self-defence," striking the driver, who was transported to a hospital where he died of his injuries, ICE said.
Reuters could not immediately verify the man's immigration status or the circumstances of the shooting.
Video footage captured by a surveillance camera from a nearby business and reviewed by Reuters showed a person lying on the ground beside a white van and surrounded by officers, in what appeared to be the aftermath of the shooting.
Houston Fire Department spokesperson Rustin Rawlings said firefighters dispatched to the scene found a man who was shot in the abdomen undergoing cardiopulmonary resuscitation before he was taken to a hospital.
In several instances over the past year, initial statements from immigration enforcement agencies about their use of force have been challenged by video footage or other evidence.
In October, a Chicago-area woman, Marimar Martinez, was accused of ramming law enforcement officers with her car.
She was shot five times but survived. Charges against her were ultimately dropped and video evidence suggested that the agents could have struck her car themselves.
Trump administration officials also said that two US citizens shot dead by federal agents on the streets of Minneapolis in January, Renee Good and Alex Pretti, had threatened bodily harm to the agents before they were killed, despite video evidence to the contrary.
In May, a Minnesota prosecutor charged an ICE agent with assaulting a Venezuelan man in a non-fatal shooting in Minneapolis during the same immigration crackdown in which Good and Pretti were killed.
ICE said in its statement on Tuesday that the FBI would lead the investigation into the shooting.
The deadly confrontation in Houston came amid a recent increase in the number of ICE arrests nationwide, with immigration officers picking up around 2000 migrants a day last week, according to two people familiar with the matter.
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