The 19-year-old's parents, Charlotte Charles and Tim Dunn, said they hoped the meeting, due to take place on Tuesday evening, would prove a "positive development" in their fight for justice.
Their teenage son was killed when his motorbike was involved in a head-on collision outside RAF Croughton in Northamptonshire in August.
Anne Sacoolas, the motorist allegedly responsible for the crash, was given diplomatic immunity and allowed to return to the US after the crash.
The family travelled to America earlier this week, and said they would be trying to convince President Donald Trump to reverse the decision.
In a post on the Justice 4 Harry GoFundMe page, Ms Charles and Mr Dunn said of the meeting at the White House: "We are grateful for the invitation, which we hope represents a positive development in our fight for justice.
"Our priority, as any parent will understand, is justice for our child. We believe this can only be achieved if Anne Sacoolas returns to England and engages properly with the justice system, where she will be treated fairly in a proper investigation of what happened to our son on that day - an investigation that cannot happen without her co-operation.
"Friends tell each other the truth. If Britain and America are friends then we believe there should be no possibility of a citizen of one country hiding from justice in another while falsely claiming a privilege such as diplomatic immunity."
The family, who met Foreign Secretary Dominic Rabb in the UK last week, have also demanded an investigation over the Foreign Office's (FCO) advice to Northamptonshire Police that Ms Sacoolas had diplomatic immunity.
They are demanding to see all emails, messages and notes sent in relation to her immunity status.
Speaking in New York, family spokesman Radd Seiger said: "We want to conduct an investigation into the FCO's decision to advise Northamptonshire Police that this lady had the benefit of diplomatic immunity.
"What we don't know is whether somebody cocked up or whether they were put under pressure by the Americans to concede."
If they are not happy with the documents, the family say they will then ask for a judge's opinion on the lawfulness of the FCO's decision.
Mr Seiger added: "If we're not satisfied, then we'll go to a judicial review and ask a High Court judge to review it all."
On Saturday, before the family left for the US, they received a letter from Mr Raab, saying that Mrs Sacoolas, 42, no longer had immunity.
"The letter is worded very carefully, they're not saying it's just a change of heart - they are saying that it's an evolution," said Mr Seiger.
He also said that the family's lawyers, Mark Stephens and Geoffrey Robertson QC, believe "they've (the FCO) been lying to you".
Ms Charles and Mr Dunn said: "We were astonished last Saturday to receive the hand-delivered letter from the FCO indicating that Mrs Sacoolas did not in fact have diplomatic immunity.
"We were still reeling from our terrible meeting with Dominic Raab, Foreign Secretary, last Wednesday, just three days earlier, where he was adamant that Mrs Sacoolas did have immunity.
"Something is not right. We will not rest until we get the truth. We will not let Harry passing be in vain."
Mrs Sacoolas has said she is "terribly sorry" about the incident and that she had "no time to react" when she saw Mr Dunn's motorbike.
Her lawyer, Amy Jeffress, told the Mirror: "Anne was driving on the wrong side of the road and is terribly, terribly sorry for that tragic mistake.
"She wants to meet the family and apologise and take responsibility. Anne had no time to react when she saw the motorbike."
Ms Jeffress, from US firm Arnold & Porter, said the diplomat's wife also stayed at the scene and flagged down other people to help the teenager.
Ms Jeffress added: "She spoke to Harry to tell him that she would call for help.
"She waved down another car. That driver offered to assist Harry so that Anne could comfort her young children in her car."
The FCO has been contacted for comment.