The US president made the comment during a radio interview just three months after Diana was killed in a car crash in Paris, the Washington Post reported.
The audio from the November 1997 interview between the US president and DJ Howard Stern has been leaked by website Factbase.
In the clip, Mr Trump spoke about the possibility of having sex with the princess, saying he would have ‘nailed’ her.
He also joked about making her have a HIV test.
In the interview, Mr Stern asks: ‘Why do people think it’s egotistical of you to say you could’ve gotten with Lady Di?
‘You could’ve gotten her, right? You could’ve nailed her.’
Mr Trump replied: ‘I think I could have.’
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Mr Stern then imagined a conversation between Mr Trump and Diana, saying: ‘Hey Lady Di, would you go to the doctor?’
Mr Trump then joked: ‘Go back over to my Lexington Avenue doctor. We wanna give you a little check-up.’
The two men had previously joked that Mr Trump would make sure women had HIV tests before sleeping with them.
In the same interview, Mr Trump spoke about Diana, commenting on her appearance.
He said: ‘She had times, you know, it was interesting, she had times when she didn’t look great, and sometimes you look better than anybody in the world. But she was supermodel.”
In another interview with Mr Stern, in 2000, Mr Trump said he would have slept with Diana ‘without even hesitation’.
He said: ‘Lady Di was truly a woman with great beauty. She was supermodel beautiful; she had the height, she had the beauty, she had the skin, the whole thing.
‘She was crazy, but you know these are minor details.’
In 2015, broadcaster Selina Scott claimed Mr Trump was obsessed with the princess and ‘bombarded her with flowers’, picturing Diana as ‘the ultimate trophy wife’.
In his 1997 book, The Art of the Comeback, Mr Trump wrote: ‘I only have one regret in the women department – that I never had the opportunity to court Lady Diana Spencer.
‘I met her on a number of occasions. I couldn’t help but notice how she moved people. She lit up the room with her charm, her presence. She was a genuine princess – a dream lady.’
(Main picture: PA)