The move came as a Minnesota court allowed journalists and members of the public, by appointment only, to view body-camera footage from Mr Floyd's arrest.
The family of George Floyd, whose murder by a Minneapolis police sparked worldwide protests, announced Wednesday that they are suing the city for his death.
Attorney Benjamin Crump said the family filed the wrongful death lawsuit in the Federal District Court in Minneapolis, even though the four officers involved in Mr Floyd's death have yet to go on trial on murder and related charges.
Mr Floyd died on 25 May when now-fired officer Derek Chauvin held his knee to his neck for more than eight minutes after the 46-year-old black man was detained for allegedly passing a counterfeit bill at a grocery store.
Much of the country was stunned by a bystander video which showed Mr Floyd pleading for his life, saying "I can't breathe," before he went limp and died.
Mr Floyd's death touched off riots and demonstrations in Minneapolis, neighbouring Saint Paul and across the country over police mistreatment of African-Americans.
It also kicked off protests against racism and police brutality in other countries across the world, including Australia.
"It was not just the knee of Officer Derek Chauvin on George Floyd's neck for eight minutes and 46 seconds, but it was the knee of the entire Minneapolis Police Department on the neck of George Floyd that killed him," Mr Crump said in front of the federal courthouse in Minneapolis.
Mr Floyd's family did not say how much in damages they are asking.
But the civil lawsuit has the potential to bring them millions of dollars if they can prove that the city's policies are at fault for the behaviour of Mr Chauvin and the other officers involved.
In May 2019, Minneapolis was ordered to pay $20 million ($28.5 million AUD) to the family of a yoga teacher killed by a police officer as she sat in her car.
"The city of Minneapolis has a history of policies and procedures and deliberate indifference when it comes to the treatment of arrestees, especially black men, that cries out for training and discipline," Mr Crump told reporters.
"This is an unprecedented case, and with this lawsuit, we seek to set a precedent to make it financially prohibitive that the police won't wrongfully kill marginalised people, especially black people, in the future."
Mr Crump called Mr Floyd's case a "tipping point for policing in America."
"While all of America is dealing with the public health crisis of the coronavirus epidemic, black America has to deal with another public health pandemic of police brutality," he said.
Mr Chauvin, 44, remained at a maximum-security prison in Stillwater, Minnesota.
Three other fired officers who either helped hold Mr Floyd down or stood guard are charged with abetting second-degree murder and manslaughter.
Body camera footage
Meanwhile, a court has allowed journalists and some members of the public, by appointment only, to view body-camera footage from two of the officers involved in Mr Floyd's arrest.
Judge Peter Cahill, without explanation, has declined to allow publication of the video.
The footage shows the officers' view of a death already widely seen on a bystander's mobile phone video.
"I'm not that kind of guy," Mr Floyd says as he struggles against the officers.
"I just had COVID, man, I don't want to go back to that." An onlooker pleads with Mr Floyd to stop struggling, saying, "You can't win!". Mr Floyd replies, "I don't want to win!"
A few minutes later, with Mr Floyd now face-down on the street, the cameras record his fading voice, still occasionally saying, "I can't breathe" before he goes still.
The recordings from officers Thomas Lane and J Kueng are part of the criminal case against them and two other officers in Mr Floyd's death.