Downing Street confirmed Mr Johnson had broached the subject in an earlier telephone conversation with the US president, asking for British police to be allowed to pursue the young motorcyclist’s death.
Mr Trump appeared to defend the allegation that the car was driving on the wrong side of the road at a White House press conference on Wednesday.
When asked about the diplomatic immunity row by reporters, Mr Trump called the incident a “terrible accident” but suggested driving on the opposite side of road was confusing – saying “it happens”.
“The woman was driving on the wrong side of the road, and that can happen. You know, those are the opposite roads, that happens,” he said.
“I won’t say it ever happened to me, but it did.
“When you get used to driving on our system and then you’re all of a sudden on the other system where you’re driving – it happens. You have to be careful, very careful.”
Harry, 19, was killed when his motorbike crashed into a car on August 27.
The suspect, 42-year-old Anne Sacoolas, who is reportedly married to a US intelligence official, was granted diplomatic immunity following the crash.
The car was thought to have been driving on the wrong side of the road after leaving RAF Croughton in Northamptonshire – a military base used by the US Air Force.
Mr Trump confirmed that his administration would seek to speak to the driver, after Number 10 said the Prime Minister had urged him to “reconsider” in an earlier phone call.
“The Prime Minister urged the President to reconsider the US position so the individual involved can return to the UK, co-operate with police and allow Harry’s family to receive justice,” said a Downing Street spokesman.
“The leaders agreed to work together to find a way forward as soon as possible.”
The comments came after Harry’s family said a meeting with Foreign Secretary Dominic Raab over the death of their son felt like a “publicity stunt”.
Mr Raab met Harry’s mother Charlotte Charles and father Tim Dunn on Wednesday afternoon after having talks with US Ambassador Woody Johnson on Tuesday.
Harry’s mother told reporters she felt “let down by both governments”.
Ms Charles said: “I can’t really see the point as to why we were invited to see Dominic Raab. We are no further forward than where we were this time last week.
“Part of me is feeling like it was just a publicity stunt on the UK Government side to show they are trying to help.”
Speaking after the meeting, Mr Raab said he shared the “frustration” felt by the family and vowed to continue to “press the US authorities” into co-operating with the UK investigation.
“I share the frustration of Harry’s mother and father,” said the Secretary of State.
“They have lost their son and the justice process is not being allowed to properly run its course.
“The Prime Minster has spoken to President Trump this evening about the case and made clear that what has happened is not acceptable.
“We are continuing to press the US authorities for their co-operation to ensure the police can pursue this case unimpeded and to allow Harry’s family to get justice.”
The family’s lawyer and spokesman Radd Seiger said the family would be willing to talk with Mr Trump about the issue and confirmed they plan to travel to the States.
“Meet us. Let’s have a chat. Nobody wants to litigate,” he said.
Mr Seiger said they were engaging lawyers to take a civil case against Mrs Sacoolas in America.
Asked about his son by reporters, Mr Dunn became emotional and was supported by his family as he called him a “special boy”.
He said: “He didn’t have a bad bone in his body, he just loved his family, he just loved everything.
“He was a special boy and I miss him like mad.”
Northamptonshire Police have also asked the US to consider waiving the immunity.
Prior to meeting the US Ambassador, Mr Raab raised the case in a telephone call with US Secretary of State Mike Pompeo.