Sacoolas, who is reportedly married to a USA intelligence official, left the United Kingdom after the crash claiming diplomatic immunity.
Trump has called the accident "terrible", noting that motorists in the U.S. and United Kingdom drive on opposite sides of the road.
Harry's parents Charlotte Charles and Tim Dunn were told that the woman at the centre of the police investigation into the death of their son in August was in a room next door while the meeting was taking place.
Mr Dunn's family say there is no evidence she ever had immunity because her husband is not a diplomat.
Ms Charles said: "We didn't get the answers that we wanted". They rejected the offer by the president, saying it would not be appropriate with no mediators or therapists present.
Anne Sacoolas gave every impression she would "engage" with Northamptonshire Police's inquiry into the smash that killed 19-year-old Harry Dunn outside RAF Croughton, the force's head of operations said.
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Pas forcément mais je ne peux pas me contenter de ce que j'ai aujourd'hui à Chelsea . Je reviens à Chelsea gonflé à bloc, cela fait du bien d'être décisif.
"To just have that [potential meeting] sprung on them ... it wasn't mentioned when they called me, I thought we were coming down to have a debate on the diplomatic immunity laws", Mr Seiger told BBC Breakfast. "And that's not just for us".
After the family declined the meeting with Sacoolas, Seiger said he noticed a bank of photographers, who seemed to have been positioned to capture the dramatic moment.
"We weren't ready to meet her - it would have been too rushed".
Harry's father, Tim, added that he did not feel that the meeting was a publicity stunt.
Dunn's parents took their case directly to USA audiences this week, holding a NY news conference and urging Sacoolas to return to Britain.
He added the president had appeared moved when he learned Harry was a twin, and had agreed he would be fighting for justice if one of his own children had been killed.
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Harry Dunn, 19, died on 17 August in a head-on collision between his motorcycle and a auto being driven on the wrong side of the road in Northamptonshire in central England.
Charles and Dunn have not yet met Sacoolas.
Dunn's mother Charlotte Charles asked Sacoolas to "do the right thing" and to face both the British legal system and "us, a broken family", during the press conference, according to The Associated Press.
'We believe two of them were in the auto and we can't think about what they're going by both.
"We came on Sunday and within two days we've been to the White House". We raised our youngsters to be as honest because the day is lengthy.
Trump last week called the collision "a bad accident" and said he planned to intervene by potentially arranging a meeting between the Dunns and Sacoolas. The prime minister asked the president to do all he could to help resolve this tragic issue. Unfortunately they wanted to meet with her and unfortunately when we had everybody together they decided not to meet - perhaps they had lawyers involved at that time - I don't know. He would have done the same for anyone of us. "We still want justice for Harry and we will take it as far as we possibly can to ensure that that's done".
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'He mentioned he would try to comprise at this from yet another angle.