US Vice-President JD Vance has defended the immigration agent who shot and killed a 37-year-old Minneapolis woman amid tensions between state and federal officials.

Renee Nicole Good was shot in the head in front of a family member on Wednesday, local time, during an immigration enforcement surge.

Mr Vance said the shooting was justified and that Ms Good was a "victim of left-wing ideology".

"I can believe that her death is a tragedy while also recognising that it is a tragedy of her own making," he said.

Mr Vance said the officer was clearly acting in self-defence.

There has been an outpouring of grief and anger in the community following the shooting. (AP: Mike Householder)

FBI 'reversed course' over joint probe

Minnesota's Bureau of Criminal Apprehension (BCA) said it had been blocked by the FBI from taking part in a joint investigation into the shooting.

Following the shooting, the BCA initially said it had agreed to jointly investigate the shooting with the FBI.

But on Thursday, a day after the shooting, BCA Superintendent Drew Evans said the federal agency had "reversed course" and taken sole control over the probe.

He said that step meant the state bureau would no longer have access to the scene evidence, case materials or interviews.

"As a result, the BCA has reluctantly withdrawn from the investigation," Superintendent Evans said.

The FBI and the office of US Attorney Daniel Rosen, the chief federal prosecutor in Minneapolis, did not immediately respond to questions about the BCA statement.

Keith Ellison, the state's Democratic attorney general, told CNN the FBI's decision was "deeply disturbing" and said state authorities could investigate with or without the cooperation of the federal government.

Renee Nicole Good was shot inside her car on Wednesday, local time. (Reuters: Tim Evans)

Ms Good's death has left Minneapolis on edge, with protesters taking to the streets in anger and schools cancelling classes as a precaution on Thursday.

Minnesota and Trump administration officials have offered starkly different accounts of the shooting, with US President Donald Trump describing the slain woman as a "professional agitator".

Democratic politicians and protesters have contested that claim, with the Minneapolis City Council saying she was "out caring for her neighbours" and died "at the hands of the federal government".

Vance says officer deserves gratitude

Homeland Security Secretary Kristi Noem said on Wednesday, local time, that the officer who shot Ms Good had been "dragged" by a vehicle during a previous incident in June.

According to court documents, the officer was part of a team trying to apprehend a man in the country illegally.

JD Vance defended the shooting as self-defence during a press conference on Thursday. (Reuters: Kevin Lamarque)

He broke a window and reached into the vehicle, attempting to open the door when the driver sped off, dragging the officer the length of a football field in 12 seconds.

The officer's right arm was bleeding, and an FBI agent applied a tourniquet.

He was transported to a hospital, where he received more than 50 stitches.

Prosecutors said he had "suffered multiple large cuts and abrasions to his knee, elbow, and face".

Mr Vance said the ICE officer "deserves a debt of gratitude".

"This is a guy who's actually done a very, very important job for the United States of America," he said.

"He's been assaulted. He's been attacked. He's been injured because of it."

City and state officials blame immigration surge for shooting

Federal authorities in Minneapolis used chemical agents on protesters during democrations on Thursday, a day after the shooting. (AP: Tom Baker)

The agent was among 2,000 federal officers that the Trump administration had announced it was deploying to the Minneapolis area in what the Department of Homeland Security described as the "largest DHS operation ever".

But Wednesday's shooting drew immediate condemnation from city and state officials who blamed Mr Trump's immigration enforcement surge for sowing chaos in the city's streets.

DHS officials, including Ms Noem, defended the shooting as self-defence and accused the woman of trying to ram agents in an act of "domestic terrorism".

Minneapolis Mayor Jacob Frey, a Democrat, called that assertion "bulls**t" and "garbage" based on bystander videos taken of the incident that appeared to contradict the government's account.

Videos showed two masked officers approaching Ms Good's car, which was stopped at a perpendicular angle on a Minneapolis street.

As one officer ordered Ms Good out of the car and grabbed at her door handle, the car briefly reversed and then began driving forward, turning to the right in an apparent attempt to leave the scene.

A third officer, positioned in front of her car on the left, drew his gun and fired three times while jumping back, with the last shots aimed through the driver's window after the car's bumper appeared to have cleared his body.

The video did not appear to show contact and the officer stayed on his feet, though Ms Noem said he was taken to a hospital and released.

Reuters/AP