Later Friday evening, Time disputed the president's claim, tweeting that he "is incorrect about how we choose Person of the Year".
However, the publication turned down the President's claims, saying the award victor will only be declared on December 6.
In past years, Trump has publicly derided out the magazine for not being selected.
Will Time Magazine respond to Sean Hannity?
Trump was actually named Time's "Person of the Year" last year shortly after the election.
The magazine's editors are the ones who select the Person of the Year, but the publication offers a poll where readers can vote for their pick.
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It wasn't all bad news for hard alcohol - almost 60 percent of respondents said the drink made them feel energetic and confident. It is clear from the study that people associate a range of emotions - good and bad - with drinking different types of alcohol .
But added that they had also requested for an interview and photo shoot. "You should run Time!".
The Washington Post reported in June that Time asked the Trump Organization to remove several fake covers featuring the president in his days as host of "The Apprentice" that were hanging in his golf clubs.
"They used to call it "Man of the Year" but they can't do that anymore so they call it person - they want to be politically correct, that's OK", he said at a rally in Iowa. "But I would have to agree to an interview and a major photo shoot".
Hollywood just called to say that I was PROBABLY going to be named the next James Bond but I would have to agree to an interview and a major photo shoot.
Mr Trump's professed indifference to the award contrasts a little with previous years, where he has paid close attention to the judges' decision.
But when he was honoured with the title previous year, there was a change in tone.