Harry Dunn's family are asking the government to turn over all documents it has about the diplomatic immunity status of the suspect in the teenager's death.
Anne Sacoolas, 42, left the UK just days after a road crash which killed the 19-year-old motorcyclist.
A Dunn family spokesman said if the advice was not disclosed they would launch a judicial review.
Mr Dunn, 19, died near RAF Croughton in Northamptonshire on 27 August.
Family spokesman Radd Seiger said their lawyers, Mark Stephens and Geoffrey Robertson QC, were ready to launch a full investigation into the role the Foreign and Commonwealth Office (FCO) played in the decision to grant immunity to Mrs Sacoolas.
Mr Seiger said: "What Mark [Stephens] and I are going to do, is we are going to write to the FCO very shortly, explaining that we don't want to do a judicial review, but to avoid that, please let us have the following documents - all e-mails, messages and notes in relation to your advice to Northamptonshire Police that this lady had it [diplomatic immunity].
"What we don't know is whether somebody cocked up or whether they were put under pressure by the Americans to concede.
"But we want to conduct an investigation into the FCO's decision to advise Northamptonshire Police that this lady had the benefit of diplomatic immunity.
"If we're not satisfied, then we'll go to a judicial review and ask a High Court judge to review it all."
On Monday, Harry's parents Charlotte Charles and Tim Dunn gave interviews on US TV after flying to New York in a bid to publicise their case.
They hope media exposure will put pressure on the US government to force Mrs Sacoolas to return to the UK.
Over the weekend Mrs Sacoolas - who is reportedly married to a US intelligence official who was stationed at RAF Croughton - broke her silence over Mr Dunn's death in a letter via her lawyers.
In it she said she wanted to meet his parents "so that she can express her deepest sympathies and apologies for this tragic accident".
Mrs Sacoolas was said to be covered by diplomatic immunity as the spouse of a US intelligence official, though that protection is now in dispute.
On Saturday, Foreign Secretary Dominic Raab wrote to Mr Dunn's family to explain that the British and US governments now considered Mrs Sacoolas's immunity irrelevant.
He said the matter was now "in the hands" of Northamptonshire Police and the Crown Prosecution Service.
The letter was sent three days after a meeting between the Dunn family and Mr Raab, who was described as "twitchy" by Mr Seiger.
"He [Mr Raab] was stiff, he was cold, he was unpleasant, he was rude.
Harry's parents described the meeting as "terrible" and said Mr Rabb was "adamant that Mrs Sacoolas did have immunity".
"We do not know what is going on but the matter is now in the hands of our legal team," they added.