The dossier containing the allegations was a two-page memo appended to a report on Russian interference in the 2016 election that was presented last week to Mr Trump and to President Barack Obama.
Mr Trump angrily denounced the unsubstantiated claims, attacking United States intelligence agencies over the leak of the information.
"I think it's a disgrace that information would be let out. I saw the information, I read the information, outside of that meeting. It's all fake news. It's phoney stuff. It didn't happen. And it was gotten by opponents of ours, as you know because you reported it and so did many of the other people. It was a group of opponents who got together, sick people, and they put that crap together. So I will tell you that, not within the meeting but outside of the meeting, somebody released it. It should never have been - number one, it shouldn't have even entered paper (sic), but it should never have been released."
The document was first reported by CNN, with online outlet BuzzFeed publishing the full document.
When CNN's Senior White House Correspondent attempted to ask a question, Mr Trump refused to answer, describing CNN as "terrible" and saying CNN is "fake news."
He also referred to BuzzFeed as "failing pile of garbage" and said the publication would "suffer the consequences" for publishing the unconfirmed report.
Mr Trump also used the news briefing to say it's likely Russia was behind the alleged hacking attacks of the Democratic Party.
"As far as hacking, I think it was Russia. But I think we also get hacked by other countries and other people and I can say that, you know when we lost 22 million names and everything else that was hacked recently, they didn't make a big deal about that, that was something that was extraordinary, that was probably China. We have much hacking going on."
Vladimir Zhirinovsky, leader of the Russian Liberal Democratic Party, says the claims are a complete fabrication.
"I think this is another attempt to derail relationships between Russia and the US. There is no compromising information. Trump earned an honest livelihood in the US and no one will be able to find fault with him. That is honestly earned money. He has never taken part in any dubious operations. And we do not have anything bad on him. We wish him to assume office on 20 January and are confident that he will do everything better than Barack Obama."
Donald Trump's choice for Secretary of State, Rex Tillerson, says it's a "fair assumption" Russian President Vladimir Putin knew about Moscow's involvement in America's 2016 presidential election.
Mr Tillerson has told the Senate Foreign Relations Committee that while he's not privy to detailed intelligence about Russia's hacking, he does believe Russia poses a danger in the current global environment.
"Our NATO allies right to be alarmed at a resurgent Russia. But it was in the absence of American leadership that this door was left open and unintended signals were sent. We backtracked on commitments we made to allies, we sent weak or mixed signals with red lines that turned into green lights. We did not recognise that Russia does not think like we do."
Touching on other issues, Mr Trump described how he will separate himself from his global business operations to avoid conflicts of interest, and he reiterated his commitment to bring manufacturing jobs back from overseas plants.
Mr Trump also says he will be making a decision in the next two weeks about who he plans to put up for justice of the United States Supreme Court and says a replacement for "Obamacare" will be offered with the confirmation of his health secretary.
Republicans agree on repealing the law but nearly seven years after its passage have failed to reach agreement on what to replace it with.i
