The authorities in Washington state have hit back after U.S. President Donald Trump threatened to intervene over a police-free autonomous zone protesters have set up in the western U.S. city.

Washington Governor Jay Inslee called on President Trump to "stay out of Washington state's business" in a Thursday morning tweet responding to the president's criticism of the state's handling of the George Floyd protests.

"A man who is totally incapable of governing should stay out of Washington state's business. 'Stoop' tweeting," Inslee posted, referencing a misspelling in the president's tweet.

Seattle Mayor Jenny Durkan told Donald Trump to "Go back to your bunker," a nod to reports Trump was rushed by Secret Service agents to a secure area in the White House as demonstrations against racism and police brutality sparked by the death of Floyd reached the president's residence.

Trump sparked the spat when he threatened to intervene in the neighborhood in Seattle dubbed "Capitol Hill Autonomous Zone," or CHAZ, which was agreed upon by demonstrators and the city's police department.

"Take back your city NOW. If you don't do it, I will," Trump warned Durkan and Inslee – both Democrats – in a tweet late Wednesday, calling the protesters "domestic terrorists" who have taken over Seattle.

Durkan defended the right to protest and told President Trump: "Make us all safe. Go back to your bunker."

"The threat to invade Seattle, to divide and incite violence in our city, is not only unwelcome, it would be illegal," she tweeted.

Protests have taken place across the country following the death of the unarmed black man killed in police custody in Minneapolis on May 25.

Officials in Seattle have denied reports that left-wing activists are behind the setting up of the autonomous zone. 

In the CHAZ area Thursday there were tents with supplies for volunteer medics as well as free gourmet food donated by local restaurants, along with fruit, snacks, and water bottles for the taking.

The sunny afternoon gathering had a relaxed air, with people of all ages, including mothers with children, milling around the car-free streets.

At one point a crowd locked arms and prevented two police officers from reaching a boarded up police station in the area. The officers failed to break in when they tried to enter through a different road.

(With input from AFP)