Donald Trump avoided a direct question about whether his team were in contact with Russia during his victorious presidential election campaign.
Speaking at a news conference, the President-elect dismissed the claims in a leaked intelligence dossier as “fake news”.
But he appeared more reticent when asked about his relationship with Moscow.
“Can you stand here today, once and for all and say that no one connected to you or your campaign had any contact with Russia leading up to or during the presidential campaign,” a reporter asked. "And if you do indeed believe that Russia was behind the hacking, what is your message to Vladimir Putin right now?"
In reply, Mr Trump said: "He shouldn’t be doing it. He won’t be doing it. Russia will have much greater respect for our country when I’m leading than when other people have led it."
He did not directly address the relationship between his campaign team and Russia.
Changing the subject, he complained about Chinese attempts to hack into American computers and cyber security.
Mr Trump's comments followed the publication of a series of documents which claimed that Russian intelligence operatives had developed information against him that was both salacious and compromising.
BuzzFeed published scans of the apparent documents, although stressed that they remain unverified.
The 35-page dossier, purportedly written by a British intelligence agent, alleged that Russia "has been cultivating, supporting and assisting Trump for at least five years".
Mr Trump attacked the "sick" people who put together what he called a "c**p" dossier.
While doubt has been cast on its contents, the US Director of National Intelligence James Clapper has previously confirmed that security agencies believe Russia did try to influence the US election.
He told the Senate Select Committee on Intelligence that the FBI had "high confidence" in its conclusion that Russia launched cyber attacks on the election to "denigrate" Hillary Clinton.
"We have multiple high quality sources that contribute to that assessment," he said. "Attributing cyber operations is difficult but not impossible."
