The mayor of Phoenix apologized on Sunday after videos emerged showing police in the southwestern US city violently arresting a family with young children.

The videos, shot when the arrest happened last month, show police using foul language while shouting at the family to get out of their car, and threatening to shoot them.

A woman later emerges from the vehicle with two young children. She hands them over to a bystander before being arrested. 

Separately, an officer is shown kicking the legs of another man who he is handcuffing against a police car.

"I, like many others, am sick over what I have seen in the video depicting Phoenix police interacting with a family and young children," Mayor Kate Gallego wrote in a statement posted on Twitter.

Calling the officers' actions "inappropriate" and "clearly unprofessional," she wrote "There is no situation in which this behavior is ever close to acceptable."

"I am deeply sorry for what this family went through, and I apologize to our community," she said.

In an appearance on the local ABC affiliate, Phoenix police chief Jeri Williams said the officers had responded to a report of shoplifting when they encountered the family.

"I'm sorry this incident happened," she said, adding that it was being investigated.

ABC reported that the family has filed a $10 million claim against the city, alleging civil rights violations.