Although speculation about Mr Trump’s apparent susceptibility to Russian influence has been a constant feature of media reports since his election victory, Kremlin spokesman Dmitry Peskov said there has been little meaningful communication between Washington and Moscow.
Mr Trump's campaign statements led to speculation that sanctions against Russia imposed for its interference with Ukraine might be dropped, but Mr Peskov told CNN: “Unfortunately, we don’t have a better understanding of when this dialogue can begin.
“Russia will never initiate putting this issue on the agenda.”
The two leaders are yet to meet face to face, although they had a telephone conversation shortly after Mr Trump was inaugurated, prompting speculation that three years of tensions between the two countries might be coming to an end.
But the phone call was said to have entailed only Mr Putin offering his congratulations to Mr Trump followed by a brief discussion on the current state of relations between the former Cold War enemies.
Asked about allegations Russian state-sponsored hacking could have influenced the outcome of the US election after intelligence agencies concluded with “high confidence” such interference took place, he said: “We don’t know what’s the reason for these words. We’ve never seen any evidence, and we’ve never heard something trustful.
“What we have seen — an open, a public part — of a report by one of the agencies, special agencies of the United States. And I would humbly say that it’s not a paper of high quality, in terms of being really trustful.”
Mr Putin has called Mr Trump “a very outstanding man” who is “uniquiely talented”. In response to the compliments paid during the US presidential race, Mr Trump said: “It is always a great honour to be so nicely complimented by a man so highly respected within his own country and beyond.“
During a debate with Hillary Clinton, Mr Trump said Mr Putin had been a better leader than President Obama.
"The man has very strong control over his country.
“Now, it's a very different system, and I don't happen to like the system, but certainly in that system he's been a leader, far more than our President has been a leader.”
When asked about allegations President Putin orchestrated the deaths of his political opponents and journalists, Mr Trump defended the former KGB man in an interview with Fox News, saying: “I haven’t seen any evidence that he killed anybody.” He added that America was not “so innocent” when it came to killing people.
“Well, I respect a lot of people but that doesn't mean I'm going to get along with him. He's a leader of his country. I say it's better to get along with Russia than not.
”And if Russia helps us in the fight against Isis, which is a major fight, and Islamic terrorism all over the world, that's a good thing.
“Will I get along with him? I have no idea.”
If the two presidents are able to establish a rapport, it could pave the way for deals on Ukraine and Syria, the two major sources of friction between their countries during the administration of Barack Obama after Putin backed Assad in Syria, annexed Crimea and invaded Ukraine.
Both leaders have spoken about ending the enmity that has dragged US-Russia relations to their lowest ebb since the Cold War.