Unsubstantiated allegations say Russia has embarrassing information about him.
Without referring to the stories, the president-elect tweeted: "FAKE NEWS - A TOTAL POLITICAL WITCH HUNT!"
Mr Trump is due to hold a news conference on Wednesday, nine days before he takes office.
It was meant to be about his attempts to separate himself from his business commitments, to address concerns about conflicts of interest.
But it now comes at an awkward time for the Republican president-elect.
As well as the allegations of a personal nature, the media reports also allege secret communications between the Trump presidential campaign and Moscow.
The US intelligence agencies reportedly presented the information to the president-elect and his predecessor at the White House, President Barack Obama.
They were said to be attached to a wider classified report into alleged interference by Russia into the US election.
But there has been nothing said publicly by the US intelligence community to support these unsubstantiated claims.
In recent weeks, Mr Trump has been under pressure to concur with the findings of US spy agencies who say Russia was behind the hacking of Democratic Party emails during the presidential campaign.
They say the order came from the Kremlin to sway the election for Mr Trump and away from his Democratic opponent, Hillary Clinton.
But so far he has failed to explicitly agree with the conclusions of the intelligence services.
And he has condemned those who oppose good relations with Russia as "fools".
Russia has denied any involvement in the hacks and accused the US of conducting a witch hunt.
Sources: New York Times, CNN
Donald Trump has decried as a "political witch hunt" US media reports that Russian intelligence agencies have obtained personally compromising material related to him.