In the footage, the man, believed to be in his 40s, is heard groaning and repeatedly saying "I can't breathe" to the white officer.
The Minneapolis Police Department said it was responding to a reported crime.
The incident echoed that of Eric Garner, a black man who died being arrested in 2014.
Garner was placed in a chokehold and uttering the words "I can't breathe" nearly a dozen times.
The phrase became a rallying cry for activists protesting alleged police brutality against people of colour in the US.