Minneapolis policeman Derek Chauvin has been charged with third-degree murder, but lawyer Benjamin Crump told CBS news it was a case of first-degree murder.
"We think that he had intent... almost nine minutes he kept his knee in a man's neck that was begging and pleading for breath," he said.
Looting is reported in Philadelphia.
Video from two Philadelphia TV stations on Sunday shows youths smashing several police cars and looting at least one store.
Reporting on the violence in West Philadelphia, the Philadelphia Inquirer says police cars were also set ablaze.
Several US cities have imposed curfews.
President Donald Trump tweeted: "Law & Order in Philadelphia, NOW! They are looting stores. Call in our great National Guard".
The Floyd case in Minneapolis has reignited US anger over police killings of black Americans.
It follows the high-profile cases of Michael Brown in Ferguson, Eric Garner in New York and others that have driven the Black Lives Matter movement.
Photo source BBC/AFP Caption: Protesters kneel to pray in Lafayette Square near the White House in Washington DC