The piece includes an email apparently written by John Brockman, a literary agent, who recounts seeing the royal at the billionaire’s New York City home.
“I walked in to find [Epstein] in a sweatsuit and a British guy in a suit with suspenders [braces], getting foot massages from two young well-dressed Russian women,” Mr Brockman wrote.
“We think they’re liberal in Sweden, but its more like Northern England as opposed to Southern Europe,” he said.
“In Monaco, Albert works 12 hours a day but at 9pm, when he goes out, he does whatever he wants, and nobody cares. But, if I do it, I’m in big trouble.”
Mr Brockman wrote: “At that point I realised that the recipient of Irina’s foot massage was his Royal Highness, Prince Andrew, the Duke of York.”
The email is included in The New Republic piece about Mr Brockman’s ties to Jeffrey Epstein.
It is written by Evgeny Morozov, a former client of the agent’s.
In the piece Mr Morozov explains he is cutting ties with John Brockman over his relationship with Epstein, a convicted paedophile.
The Duke of York has also faced repeated questions over his friendship with Epstein, who died by suicide in prison earlier this month.
Buckingham Palace has issued strong denials over claims from a woman who said she was forced to have under-age sex with the duke at Epstein’s home.
Virginia Roberts alleged in court papers in Florida that she was forced to have sex with Andrew when she was 17, which is under the age of consent in the state.
She alleged she had sex with Andrew “three times, including one orgy” in London, at Epstein’s New York home, and at an “orgy” on his private island in the Caribbean.
The claims were later officially struck from the court records when a judge ruled they were unnecessary to the case, stating they were “immaterial and impertinent” to the “central claim” against Epstein
Buckingham Palace has branded the allegations “false and without any foundation”.
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Prince Andrew previously resigned from his role as UK trade envoy after being photographed with Epstein in Central Park in 2010.
In the emails revealed by the New Republic Mr Brockman said the photos emerged shortly after he was introduced to the royal.
“Indeed, a week later, on a slow news day, the cover of the [New York Post] had a full-page photo of Jeffrey and Andrew walking in Central Park under the headline: ‘The Prince and the Perv.
“(That was the end of Andrew’s role [as] the UK trade ambassador.),” he wrote.
Buckingham Palace has said that “any suggestion of impropriety with underage minors is categorically untrue”.
“The Duke of York has been appalled by the recent reports of Jeffrey Epstein’s alleged crimes,” a Buckingham Palace spokesperson said earlier this week.
“His Royal Highness deplores the exploitation of any human being and the suggestion he would condone, participate in or encourage any such behaviour is abhorrent.”
The Independent has attempted to speak to John Brockman but was unable to contact him.
Buckingham Palace, when contacted, said it was not commenting on the new claims but referred The Independent to earlier statements previously released and repeated above.