In a recording obtained by the paper, a voice similar to Mr Trump's purporting to be a media representative advocates for the billionaire businessman.
Reporters would hear from the "spokesman", John Miller or John Baron, when asking to interview Mr Trump.
Mr Trump denied the story to the Today Show on Friday.
"No, I don't know anything about it. You're telling me about it for the first time and it doesn't sound like my voice at all," he said.
"I have many, many people that are trying to imitate my voice and you can imagine that. This sounds like one of these scams, one of the many scams. It doesn't sound like me.
"It was not me on the phone. And it doesn't sound like me on the phone, I will tell you that. It was not me on the phone."
Donald Trump is an unusual man who often behaves, shall we say, unconventionally. There's plenty of evidence for this, whether or not he made calls 25 years ago pretending to be his own publicist.
Many New York reporters are convinced that this is a bona fide recording of the billionaire, and the fact that he's (allegedly) boasting about his dating prowess and celebrity socialising probably isn't the kind of story Mr Trump wants circulating as he tries to present himself as a more serious general election candidate.
This is just the start, however. While Mr Trump has been in the public eye for decades, and New York tabloid scrutiny can be daunting, nothing compares to the glare of the presidential spotlight. Given that this is Mr Trump's first foray into public office, his past will receive extra scrutiny - with more revelations to come. The Washington Post, for instance, has assigned a team of 20 reporters to sift through Mr Trump's background, from his business dealings to oddball stories.
This latest episode definitely falls into the latter category.
However, in 1990, Mr Trump admitted in court he and one of his employees used the name "John Baron" in business dealings during a case about undocumented Polish workers constructing Trump Tower.
A lawyer for the workers said he received a call from someone named "Mr Baron" who threatened to sue if he did not drop a lawsuit over withheld pay for the workers, the New York Times reported.
A reporter for People magazine contacted Mr Trump's office in 1991 to interview him about the end of his marriage to Ivana Trump and his relationship with model Marla Maples.
The "media spokesman" called the reporter, Sue Carswell, back and began telling her about why Mr Trump broke up with Ms Maples for Italian model Carla Bruni.
"He really didn't want to make a commitment," said the man claiming to be John Miller. "He's coming out of a marriage, and he's starting to do tremendously well financially."
Once, Mr Miller slipped out of talking about Mr Trump in the third person in a conversation with Ms Carswell, but quickly corrected himself.
Mr Trump is the presumptive Republican nominee after pushing out more than a dozen Republican rivals in the race for the White House.
Traditional Republicans are warming to him to unify the party and beat likely Democratic nominee Hillary Clinton.
The divisive candidate, who in the past has been a registered Democrat, has no prior political experience. He has made controversial calls to build a border wall with Mexico, deport undocumented immigrants from the country and ban Muslims from the US.
House Speaker Paul Ryan has still not officially backed him but the two recently met in Washington and had a "productive meeting".
Presumptive Republican presidential nominee Donald Trump masqueraded as a spokesman for himself in the 1990s, the Washington Post reports.