The teenager died when his motorbike crashed into a car outside RAF Croughton in Northamptonshire on 27 August.
The suspect, 42-year-old Anne Sacoolas - who is reportedly married to a US intelligence official - was granted diplomatic immunity following the crash.
But the family’s spokesman Radd Seiger said the Foreign Secretary Dominic Raab has written to Mr Dunn’s family about Mrs Sacoolas, saying: “The US have now informed us that they too consider that immunity is no longer pertinent.”
The letter, sent by Dominic Raab to the family, said: “We have pressed strongly for a waiver of immunity, so that justice can be done... Whilst the US government has steadfastly declined to give that waiver, that is not the end of the matter.
“We have looked at this very carefully... the UK Government’s position is that immunity, and therefore any question of waiver, is no longer relevant in Mrs Sacoolas’ case, because she has returned home.”
He added that the matter was now “in the hands” of Northamptonshire Police and the CPS.
An FCO spokesman said the office “would not be commenting further on the content of the letter”.
Before the letter was sent by the FCO, the family’s lawyer Mark Stephens said: “There are approximately 20,000 official diplomats in this country - there’s a definitive list of who is and who isn’t.
“We know definitively that this guy was not a diplomat and therefore was not entitled to diplomatic immunity. That has a number of consequences. That means that the Americans have made a false claim. She would not have been entitled to claim diplomatic immunity.”
“Anne is devastated"
Meanwhile, Mrs Sacoolas’s legal representative Amy Jeffress, from the law firm Arnold and Porter, said: “Anne is devastated by this tragic accident.
“No loss compares to the death of a child and Anne extends her deepest sympathy to Harry Dunn’s family.”
Mr Seiger said in a statement that he had spoken with Ms Jeffress and the pair had agreed “to get together asap... to discuss how we are going to achieve a solution”. He added he was studying the FCO letter “with legal and political experts” to “fully understand where that leaves us”.
He added: “That all said, clearly a positive step forward but we won’t rest until we get justice for Harry.”
It comes as Mr Dunn’s parents Charlotte Charles and Tim Dunn said they would be flying to the US to “put pressure on the US administration to do the right thing”.