Lawyer Kate Lincoln-Goldfinch introduced the man who phoned into a news conference she was holding in Austin as her client, Wilber Rafael Garces Perez.
He said he was bleeding and was not receiving adequate medical care in a Texas detention facility.
"I'm not doing well because of the gunshot I received," Perez, 28, said in Spanish as reporters listened to a speakerphone, adding that he had to sleep on the floor at the detention centre.
Lincoln-Goldfinch said at the news conference that Perez was being held at the South Texas ICE Processing Center, southwest of Austin, and he was undergoing questioning by agents.
Perez said he was released within four hours of arriving at a hospital on Sunday, and that a doctor treating him told him ICE agents were in a hurry to transfer him to a detention centre.
He said that he had been working as a DoorDash food delivery driver on Sunday when his car was sideswiped by the agents, who he said did not identify themselves or turn on any sort of lights or siren.
He said an agent shot him while he was still in his car and there had been no foot chase.
Austin police said on Sunday that he had run from agents.
Neither ICE nor the Department of Homeland Security, which oversees ICE, responded to requests for comment on Monday.
DHS on Sunday released a written statement saying that Perez was in the US illegally and that he had a final order of removal from the country. They did not provide details on the circumstance of the shooting.
The New York Times, citing two unnamed people it said were familiar with the federal investigation into the shooting, reported on Monday that ICE agents were out in Austin checking licence plates when they discovered that Perez had a deportation order.
The agents, the newspaper reported, pulled Perez over, identified themselves and said he was going to be arrested, at which point Perez allegedly fled in his car, leading to a short chase, sideswiping and shooting.
Lincoln-Goldfinch said a judge issued a deportation order for Perez after he missed a hearing in immigration court because the hearing notice was sent to a former address.
"Not that being out of status would justify any of this at all," Lincoln-Goldfinch said.
Lincoln-Goldfinch, speaking on CNN, said that account of the events "doesn't match" what Perez told her, though she provided no details on if he had been stopped by ICE agents before being shot.
Federal immigration officials have been involved in multiple shootings this year, including the killing of two US citizens in Minnesota, which sparked a widespread backlash.
More than 50 people have died in ICE detention since Trump returned to office last year.
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