Harry Dunn's family will visit the White House later for an "urgent meeting", the family's spokesman has said.
Harry, 19, died when his motorbike collided with a car outside an RAF base in Northamptonshire in August.
Anne Sacoolas, the woman involved in the crash, returned to America after being granted diplomatic immunity because of her husband's work.
Harry's parents arrived in New York on Sunday hoping to meet Ms Sacoolas – but only if she also agrees to return to the UK.
Ahead of the White House meeting, Harry's father Tim Dunn told Sky News: "I'm just still shocked that we got this invitation so quick, out of the blue we are talking to you guys then we are coming to Washington.
"Hopefully it's good news that she's [Ms Sacoolas] coming back to the UK and the government has said this is the way to go."
He added: "We came to America to get our point across and Anne back to the UK. If are going to the White House, surely this is a good step forward."
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Harry's mother, Charlotte Charles, said she was "really pleased" about the White House meeting but doesn't want to get her hopes up.
"We are keeping our fingers crossed that we're not going to go and have a 'there-there' pat on the back, but of course I'm pleased," she told Sky News.
"I just don't know what to expect and we keep our fingers crossed that it's not just an invitation to have a general chat.
"We got hopeful when we got the invite to see Dominic Raab, but we don't want to get our hopes up too much."
Speaking at a news conference on Monday, Mr Dunn called on President Trump to help secure justice for his son.
He said: "On that night there was an accident, a lady made a mistake, she killed our son, she didn't mean to kill him, she didn't mean to have an accident, but you cannot walk away from that and just leave and expect nothing to happen.
"Our boy died and he deserves to have some justice. That's all we want.
"I would say to him (President Trump) as a man, as a father, how could you let this happen, if you are a father and your child died surely you'd want that person to own up and take responsibility for their action?"
Harry's mother said all the family wanted was for Ms Sacoolas to return to the UK to face justice, saying it was "the right thing to do".
The Foreign Office previously said she no longer has diplomatic immunity because she had returned home, meaning the case could be taken forward by Northamptonshire Police.
Ms Sacoolas, 42, said in a statement from her lawyers that she is "devastated", though her whereabouts are uncertain.
Earlier on Tuesday, police said they had visited the scene where Harry was killed and that prosecutors would receive a file on the case "very soon".
Police have said they were not given advance warning that Ms Sacoolas was leaving the country three weeks after the crash.