The president told reporters his team will speak to Anne Sacoolas "very shortly" and discuss whether they can set up a meeting with the family of Harry Dunn, who was killed in a road accident in Northamptonshire on 27 August.
Police think Mrs Sacoolas pulled out on the wrong side of the road as she emerged from US spy base RAF Croughton, and collided with the 19-year-old as he rode his motorbike - causing fatal injuries.
Mrs Sacoolas, 42, was granted diplomatic immunity and returned to the US after the crash.
Speaking at the White House after a phone call with Prime Minister Boris Johnson, Mr Trump told reporters he wanted to explore the possibility "to do some healing".
He said: "We're going to speak to her very shortly and see if we can do something where they can meet. We're going to speak to her and see what we can come up with to do some healing."
Speaking to Sky News, Harry's mother Charlotte Charles said Mr Trump should do more.
"She (Mrs Sacoolas) should not have walked away from that and fled and he has the power to send her back and face what she's done," she said
"She's not setting a good example to her own children by 'running away'."
Ms Charles and Harry's dad, Tim Dunn, have repeatedly said they want to see Mrs Sacoolas return from the US, and have expressed their "anger" over how the death of their son has been handled by the UK government.
After the phone call between Mr Trump and Mr Johnson, Downing Street said the prime minister "urged the president to reconsider the US position".
Mr Trump was said to be "fully aware of the case" and "deeply saddened by what has happened", with both leaders agreeing to "work together to find a way forward as soon as possible".
The call came after Ms Charles and Mr Dunn met Dominic Raab and were left "disappointed" by their discussions with the foreign secretary.
Family spokesman Radd Seiger told reporters afterwards: "To say we are disappointed with the outcome of the meeting would be an understatement."
The foreign secretary is said to have relayed the US position on the issue, which is that the White House will not consider granting any waivers of diplomatic immunity.
Mr Seiger told Sky News the family felt the meeting was poorly handled, and that it had left them feeling "perplexed, angry and confused".
The family later said they felt the meeting had been a "publicity stunt" and they were "disgusted".
Mr Seiger also revealed that the family plans to recruit lawyers in the UK and the US with a view to possibly filing a civil claim against Mrs Sacoolas.
Ms Charles has also made a direct appeal to Mrs Sacoolas to return, telling reporters: "Do the humane thing, get on the plane and come back - from one parent to another.
"How would a human make a decision to get on a plane and run away from what she's done and try and continue her life, what kind of example is she setting to her children? She's being dishonest by running away from us."
Mr Dunn also spoke out after the meeting at the Foreign Office, and expressed doubt as to whether the phone call between the prime minister and the president would lead to anything.
He said he was "deeply disappointed" by the way the situation had been handled by the Foreign Office.
He added: "I don't think anything will come of it. I don't think the government or the Commonwealth Office have any clout to do anything. I don't think Boris will do any good, either."
Despite efforts since then to have Mrs Sacoolas brought back, the Foreign Office has dismissed doubts as to whether she should have been eligible for diplomatic immunity.
Questions were raised by an international lawyer, who told Sky News that her husband Jonathan Sacoolas does not appear on a list of registered diplomats.
The Foreign Office later insisted that he was an accredited diplomat in the eyes of the British government, and therefore his wife was entitled to diplomatic immunity.