Between meetings on Capitol Hill, the Floyds met with Biden and Vice President Kamala Harris at the White House behind closed doors, in order to have a "private discussion" with the family because of Biden's "personal relationship," with them, White House press secretary Jen Psaki said Tuesday.
Biden, she said, would put out a statement but he did not make formal remarks, as could have been expected if he had been able to sign a policing reform bill, as he had called on to happen by the anniversary, presumably with the Floyd family on hand.
Departing the White House for Wilmington Tuesday evening after meeting with Floyd's family for more than an hour, Biden said he was "optimistic" lawmakers would reach an agreement on policing reform sometime after Memorial Day.
Biden also said Floyd's 7-year-old daughter Gianna, who famously said her "daddy changed the world," was treated to ice cream and Cheetos during the Oval Office meeting and that he showed the family around the White House on a day he said "takes a lot of courage to get through."
Asked earlier whether Biden, by not making formal remarks, was missing an opportunity to use his bully pulpit to put pressure on Congress to reach a deal, Psaki defended the president's choice.
"We may just have a disagreement in terms of what the right strategic approach is to these negotiations moving forward and getting to the final outcome, which we all want to see, which is a bill that the president can sign into law," Psaki said, adding Biden remains "encouraged" by bipartisan negotiations and but she refused to offer a new deadline.
While Psaki fielded questions at the White House, Biden tweeted that the country is facing an "inflection point" on policing reform and must act.
In his post-meeting statement, Biden didn't specify a new timeline for getting policing reform passed, saying only he hopes lawmakers get him a bill "quickly."
Outside the White House, attorney Benjamin Crump said Biden told the Floyd family he's not looking for a "rushed" reform bill.
"He said that he doesn't want to sign a bill that doesn't have substance and meaning, so he is going to be patient, to make sure it's the right bill," Crump said.
George Floyd's brother, Philonise, called on Congress to pass the George Floyd Justice in Policing Act.
"If you can make federal laws to protect the bird which is the bald eagle, then you can make federal laws to protect people of color," he told reporters.
Gianna then led the family, with fists raised, in a call and response chant. "Say his name!" she said. "George Floyd!" they responded.
Floyd's family met first with House Speaker Nancy Pelosi and Rep. Karen Bass, D-Calif., chief author of the George Floyd Justice in Policing Act, Tuesday morning ahead of afternoon meetings with bipartisan senators.
Pelosi said Gianna's "prediction is coming true."
Bass acknowledged the anniversary will pass without new legislation but, surrounded by the Floyd family for a photo op, said getting it right was more important than a deadline.
"What is important is, is that when it reaches President Biden's desk it's a substantive piece of legislation, and that is far more important than a specific date. We will work until we get the job done," Bass said, before Philonise Floyd, put pressure on the Senate.
"We need meaningful legislation," Floyd said. "We need Joe Biden, we need the Senate to get this taken care of because just like Gianna said -- you're daddy's going to do what?"
"Change the world," said Gianna, as everyone applauded.
Senate Majority Leader Chuck Schumer, speaking on the Senate floor Tuesday morning, nodded to the social justice movement born out of Floyd's death.
"This was not only a fight for justice for one man and his family, who I've had the privilege to meet with, but a fight against the discrimination that Black men and women suffer at the hands of state power, not just here in America, but around the globe," Schumer said. "It is a fight that continues today."
But it is a fight that has produced no real changes over the last year -- despite widespread calls on both sides of the aisle to revisit policing in the U.S.
Lawmakers told Floyd's family what they have said in the past few days as the anniversary got closer -- that they are hopeful differences can be resolved soon.
Biden, who has acknowledged Black voters helped win the presidency, has not been especially forceful in putting public pressure on members to pass the George Floyd Justice in Policing Act since calling on Congress during his address to a joint session to "find a consensus" by the one-year anniversary.
The legislation, which passed the House on party lines in March, has yet to be voted on in the Senate, stuck in closed-door negotiations over the issue of eliminating qualified immunity, which makes it hard to sue individual police officers.
The Floyd family returned to the Hill later on to meet separately with Sen. Cory Booker, D-N.J., and Sen. Tim Scott, R-S.C., who promised the family that legislation would be named after Floyd, Crump said.
While the current legislation would need support from at least 10 Republicans to pass, Bass, Booker and Scott have working for weeks to bridge the divide between the House-passed Floyd bill and a Republican-backed proposal Scott put forward in the wake of Floyd's death that Democrats argued didn't go far enough. The trio has given no indication of what the final legislation will look like or a timeline of when it might be ready for consideration by the Senate.
Booker, one of the bill's lead negotiators, said in an interview with ABC News Live on Tuesday morning that their talks would take "weeks," at least.
"There's still work to do. That's going to take weeks ahead," Booker said, later adding, "Everything is on the table right now."
Floyd died one year ago after former Minneapolis police officer Derek Chauvin, who was found guilty on all counts for Floyd's death in April, pinned his knee to Floyd's neck for more than nine minutes -- sparking an unprecedented show of protests during a global pandemic and election year. Biden made addressing racial equity a center point of his campaign, calling it one of the four crises facing the country and pledging to implement real reform under his administration.
Two weeks after Floyd's death, his brother, Philonise, testified before House lawmakers in an emotional hearing on policing reform, pleading with them to take action.
"I'm here today to ask you to make it stop. Stop the pain. Stop us from being tired," he said.
Floyd family members were poised to end their day at Black Lives Matter Plaza near the White House -- the same site where thousands poured into the street last year demanding change in Floyd's name.
ABC News' Allison Pecorin, Molly Nagle and Sarah Kolinovsky contributed to this report.