Sky Roberts, Giuffre's brother, and Amanda Roberts, Giuffre's sister-in-law, had "strongly" urged Charles to meet with them - as well as survivors of pedophile financier Jeffrey Epstein - earlier this month.
Charles and the Queen are due to arrive in the US on Monday for a four-day visit during which they will meet President Donald Trump.
They will also be guests of honour at a White House banquet, and Charles has been invited to address a joint meeting of the US Congress.
"We need the King of England to stand up and show his unity with survivors," Sky Roberts, speaking to BBC Newsnight, said.
"We will be 10 minutes from him. We will be lobbying that same day in the same buildings that he will be entering in and out of.
"And all we ask is for a 10-minute meeting with the King to show him that we're real people, with real feelings."
On Friday, a BBC investigation revealed Epstein continued to house women in London flats for years after the Metropolitan Police decided not to investigate Giuffre's original allegation in 2015 - when she told the force she had been trafficked to the UK by Epstein in the early 2000s.
"This has been systematic failure from not only the FBI, but the Metropolitan Police there in the UK," Roberts said.
"I mean if they would have listened to Virginia and took her claim seriously the first time, it could have avoided so many other years of abuse that had occurred right under the noses of these officers."
The Roberts family also vowed to join Lisa Phillips - another Epstein survivor - in her call for a public inquiry into allegations made by women in the UK against the paedophile financier.
"We would 100 per cent call for an inquiry," Amanda Roberts said.
"Clearly, there is more things to be investigated and there's more evidence sitting there, especially if it was just discovered that, you know, he (Epstein) had been having flats years after the first reports came forward."
Earlier this week, controversy erupted in Congress following reports that some Republican members of an influential committee investigating Epstein were open to Ghislaine Maxwell being granted clemency in return for her giving evidence before them.
This has been seized on by Democrat opponents, who have condemned it as "sick" and branded Maxwell a "monster".
The British socialite is serving a 20-year prison sentence in the US for luring young girls for the pedophile financier to sexually abuse.
Asked if a pardon for Maxwell in exchange for evidence is something they could support, Amanda Roberts replied: "Never. Never.
"Even the idea of pardon and blame Maxwell in the same sentence makes my stomach completely sick - and the idea that this would be floated around as something plausible."
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