Minnesota Attorney General Keith Ellison said he was charging Chauvin, 44, with second-degree murder in addition to the third-degree murder and second-degree manslaughter charges levelled against him last week.
Former Minneapolis police officer Derek Chauvin has been charged with second-degree murder.
The other three former officers on the scene of Floyd's murder have also been charged.
Roxie, who took part in a press conference with her daughter had said that George "loved" his daughter, one of his two children.
Asked whether or not he now had the evidence to bring a first degree murder charge against Chauvin, he said: 'I wouldn't want to comment on what I have or don't have - we've charged the highest ethical charge that we can'.
The family's attorney, Benjamin Crump, standing next to Quincy, said they expected the officers to be "charged as accomplices for the killing".
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The Minnesota Attorney General said on Thursday he has not ruled out bringing first degree murder charges against Minneapolis cop Derek Chauvin for the death of George Floyd, and that he would prosecute "anything the facts allow".
The multiple charges against each officer would offer a jury more options to find them guilty.
Tens of thousands of protesters have taken to the streets to demand an end to racism and police brutality. Some had complained that the local prosecutors office took too long to arrest and charge Chauvin, giving energy to the protests in Minneapolis.
"It's just the beginning", he told Al Jazeera. If convicted, they could be sentenced to up to four decades in prison.
His dashboard camera was not activated and a report said he could have interviewed the woman while standing outside her vehicle.
"So, don't think for one minute, that we not gonna see another police-involved murder here in this state", he added, pointing to the police shooting of his "best friend" Philando Castillo, who was killed in 2016. They have continued every day since.
"We want justice", said Floyd's son, 27, who lives in Texas, standing before a large painted angel now marking the spot. Another was that police release jailed protesters.
Nationwide, more than 9000 have been arrested in connection with unrest.