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

The videos, shot by bystanders 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 behaviour is ever close to acceptable.

‘As a mother myself, seeing these children placed in such a terrifying situation is beyond upsetting,’ she said.

‘I am deeply sorry for what this family went through, and I apologize to our community.’

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.

‘Every time I look at that video it's extremely unsettling. It's not in keeping with what I expect from our employees,’ Williams told ABC15.

‘I'm sorry that this incident happened. I apologise to the family, I apologise to the community,’ she said, adding that it was being investigated.

- 'Civil rights violations' -

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

According to the claim, Dravon Ames, his fiancee Iesha Harper and their two children, aged one and four, had gone shopping at a Family Dollar store.

The couple's four-year-old daughter Island took a doll from the shop which they were not aware of until they got back in their car, according to notice of claim filed by former Arizona attorney general Thomas Horne who is representing the couple.

They drove to their babysitter's apartment when a police car pulled in behind them and an officer approached their car with his gun drawn, it says.

The claim alleges the officers ‘committed battery, unlawful imprisonment, false arrest, infliction of emotional distress and violation of civil rights.’

California Senator Kamala Harris, who is seeking the Democratic presidential nomination, said the officers' action were ‘indefensible.’

‘We want and expect law enforcement to protect and respect, not target and intimidate. We need stronger, independent police oversight and bias training, to root out and prevent abuses,’ Harris said on Twitter.

The incident comes amid heightened scrutiny of law enforcement by the public in the United States following a series of high-profile killings of African Americans.

In response, departments across the country have announced reforms such as requiring officers to wear body cameras, which Gallego said Phoenix would speed up in the wake of the controversy.

The officers involved in the incident were not wearing body cameras.

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