Anne Sacoolas - the American motorist suspected of colliding with Mr Dunn - fled back to America shortly after the incident in August.
A wife of an intelligence officer, she is said to be covered by diplomatic immunity, though that protection is now in dispute.
On Tuesday, Harry Dunn's family's spokesperson, Radd Seiger tweeted to say that they had all been invited to a meeting at the White House.
In a statement, Mr Dunn's family said they were "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 meeting comes just hours after Ms Sacoolas released a statement through her lawyer, in which she said she was "terribly, terribly sorry for that tragic mistake" and "neither she nor Harry Dunn’s family will ever be the same because of it".
Despite her apology, Mr Dunn's family called it "a cynical attempt to salvage her position and attempt to change the narrative".
Ms Sacoolas's statement goes on to say the US citizen does not dispute what happened, revealing she spoke to Harry Dunn in the moments after the crash.
"Anne was driving on the wrong side of the road and had no time to react when she saw the motorbike approaching - the crash happened too fast. Anne stayed on the scene of the accident to assist," the statement reads.
"She spoke to Harry Dunn to tell him that she would call for help.
"She waved down another car.
"That driver pulled over and offered to assist Harry so that Anne could comfort her young children, who had been in her car and were on the scene."
It states Mrs Sacoolas spoke to police at the scene and was told she should go home.
She says she spoke to police again the following day and she was told they would be in contact, but says she did not hear from them.
The statement concludes: "She and her family left the United Kingdom approximately three weeks after the accident, after they and the US authorities determined that it would be difficult for the couple and their children to remain in the small Croughton community under these tragic circumstances.
"She and her family returned home on a commercial flight.
"Our understanding is that the British authorities were informed and aware of their departure before they returned to the United States."
Earlier this week, they told a press conference in New York they felt let down by the decision to allow Ms Sacoolas to leave the UK after the crash.
In a statement, the FCO said it was continuing to ask authorities in the US for Ms Sacoolas's cooperation.
“At this tragic time for the Dunn family, the government is continuing to press the US authorities for the suspect’s cooperation and doing all that we can to support the family, police and Crown Prosecution Service in achieving justice for Harry," the FCO said.
"The individual concerned was covered by diplomatic immunity while she was in the UK under the Vienna Convention for Diplomatic Relations.
"That is why the government requested that the US waive immunity on multiple occasions.
"We deem that immunity ended when she returned to the US, and the US government has now indicated that immunity is no longer pertinent.
"The investigation rests with the police and Crown Prosecution Service.”