A PR expert consulted by the palace told ITV News the Duke's office overruled his advice not to go ahead with the interview about allegations of sex abuse.
Jason Stein says he later left his role at the Palace by mutual agreement.
Mr Stein said in a statement: "It was a mutual decision on both sides not to proceed with my employment and it was amicable.
"I made clear in the two months before that I disagreed and advised against it. I did not think it was a good idea."
In a BBC interview aired on Saturday night, the Duke of York went into detail over his friendship with convicted paedophile Jeffrey Epstein and allegations that he had had sex with a 17-year-old who claims she was trafficked into London.
In the Newsnight interview, Prince Andrew strenuously denied he had had sex with Virginia Giuffre because he had taken his daughter to Pizza Express in Woking on the alleged date.
He also claimed he had "no recollection of meeting her" or of the photo of him with his arm around Ms Giuffre being taken, suggesting it could have been doctored.
The Duke of York has faced a barrage of criticism following the interview, with the royal accused of a lack of empathy with the victims.
Andrew's attempt to set the record straight about his relationship with the convicted sex offender has been widely condemned for its unsympathetic tone and lack of remorse for the friendship.
Green Party co-leader Sian Berry said the duke’s interview for a special Saturday edition of BBC’s Newsnight programme was not “very empathetic” with the teenage girls preyed on by the disgraced financier.
Ingrid Seward, editor-in-chief of Majesty magazine, told BBC Radio 4’s World at One programme: “It was a very probing interview he’s not used to being questioned in that way and I think he seemed unable to really answer her questions with any conviction, I think myself and most other people think that the interview was a mistake.”
Ms Giuffre - then Virginia Roberts - is one of 16 women who say they were abused by paedophile Jeffrey Epstein and has previously claimed in court she was forced to have sex with Andrew.
She has previously alleged in court documents she “was forced to have sexual relations with this prince when she was a minor” in London and New York.
Later, when aged 18, she alleges they had sex again on a private Caribbean island owned by Epstein where an orgy took place.
Her allegations were struck from US civil court records in 2015 after a judge said they were “immaterial and impertinent”.
Newsnight presenter Emily Maitlis said Ms Guiffre alleged she went back to a house in Belgravia with the Duke and had sex with him on March 10, 2001, after meeting him at Tramp Nightclub in London.
But Andrew said he had "no recollection" of the event, and that there were "a number of things" wrong with the story.
"On that particular day, that we now understand is the date which is the 10th of March, I was at home," the Duke said.
"I was with the children and I'd taken Beatrice to a Pizza Express in Woking for a party at I suppose, sort of, four or five in the afternoon.
"And then because the Duchess [of York] was away, we have a simple rule in the family that when one is away, the other one is there."
ITV News Royal Editor Chris Ship on the 'extraordinary interview'
Andrew said he was on leave from the Royal Navy at the time and was therefore at home with the children.
Ms Maitlis asked why he remembered going to Pizza Express "so specifically".
The Duke replied: "Because going to Pizza Express in Woking is an unusual thing for me to do, a very unusual thing for me to do.
"I've never been... I've only been to Woking a couple of times and I remember it weirdly distinctly.
"As soon as somebody reminded me of it, I went: 'Oh yes, I remember that'."
Ms Maitlis said Mrs Guiffre had been "very specific" about Andrew that night, claiming he was "profusely sweating" and that she went on to have a bath.
But the Duke said there was a "slight problem" with that story - claiming he was unable to perspire at the time.
"I have a peculiar medical condition which is that I don't sweat or I didn't sweat at the time," Andrew claimed.
"...I didn't sweat at the time because I had suffered what I would describe as an overdose of adrenaline in the Falklands War when I was shot at and I simply... it was almost impossible for me to sweat."
He said it was only because he had "done a number of things in the recent past" that he was able to sweat again.
Leader of the Lib Dems Jo Swinson said she found it "strange" Prince Andrew did not express any understanding towards Epstein's victims during the hour-long interview.
"They're the people who should be most important and at the heart of this," Ms Swinson said."I found it strange you could have an hour-long interview without expressing any kind of empathy and understanding of the trauma they have been through."
Also in the interview, the Duke of York was questioned about his decision to stay at the home of convicted paedophile Jeffrey Epstein in 2010.
Andrew replied that it had been "convenient" and felt like the “honourable” thing to do at the time.
Epstein killed himself in prison while awaiting trial on sex-trafficking charges earlier this year.
The Duke said that with “hindsight” his decision to stay at the 66-year-old's house was “definitely the wrong thing to do” and that his judgment was “probably coloured” at the time.
However, he stated twice his relationship with Epstein had some “seriously beneficial outcomes” and it gave him the opportunity to meet people and help prepare for a future role as a trade envoy.
When asked by Maitlis if he regretted the “whole friendship with Epstein”, the Duke replied: “Now, still not and the reason being is that the people that I met and the opportunities that I was given to learn either by him or because of him were actually very useful.”
Earlier in the interview, Andrew had said he was transitioning out of the Royal Navy and wanted to the know more about the world of business as he would become the UK’s special representative for international trade and investment.
When it was pointed out during the interview that he was staying at the house of a “convicted sex offender”, Andrew said: “It was a convenient place to stay… There is… I have gone through this in my mind so many times.
“At the end of the day, with the benefit of all the hindsight one can have, it was definitely the wrong thing to do.
“But at the time I felt it was the honourable and right thing to do.
“And I admit fully that my judgment was probably coloured by my tendency to be too honourable but that is just the way it is.”
He also shows regret for not ending his friendship with Epstien sooner: "I kick myself for on a daily basis because it was not something that was becoming of a member of the royal family and we try and uphold the highest standards and practices and I let the side down, simple as that."
Andrew was photographed in 2010 walking with Epstein in New York's Central Park - two years after Epstein's conviction for soliciting a minor for prostitution.
There has been footage published showing the Duke staying at Epstein's mansion around the same time.
Buckingham Palace has branded Ms Giuffre's the allegations “false and without any foundation” and said “any suggestion of impropriety with underage minors” by the Duke was “categorically untrue”.
Prince Andrew said the whole episode had not been damaging to the Queen, only to himself, and that the wider royal family “couldn’t be more supportive”.
The Duchess of York Sarah Ferguson described the duke, who is her ex-husband, as “stoically steadfast” and a “real gentleman” in a Twitter post.