The Boston Dynamics robot would be used by the SWAT team for surveillance and LAPD officials insist it would not be equipped with weapons and would not be used for spying.
Critics worry it could be misused to harm or spy on people.
"When we think about how the LAPD, a paramilitary force that has a history of racist violence, would use a tool like a robot dog, it's probably for violence against Black and brown communities," one protester said.
The department says it can help keep officers out of harm's way.
The council heard about an hour of commentary from the public, the majority of it against the robots.
Council members decided to postpone a decision for at least 60 days.
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Critics also noted that while the New York Police Department started using a similar robot in 2020, it stopped the program after about a year because of public outcry. And the city of San Francisco last year decided against acquiring such a robot.
The value of the donation is estimated at $277,000. The device is technically referred to as a quadruped unmanned ground vehicle, but Boston Dynamics has nicknamed it "Spot."
Several council members have already showed signs of voting against the robot.
The Police Commission and the council's public safety committee have approved of the move.