Peskov also reacted to allegations of Russian meddling in US elections, dismissing an assessment by the USA intelligence community concluding with "high confidence" that Russia had interfered in the 2016 campaign in order to help Mr. Trump.
What's more, he said, America was "self-humiliating" by admitting that a foreign country had the power to intervene in its election process.
Russia's stance on this is simple, he said. "And we have never heard something trustful".
Peskov criticized what he said was "hysteria" in the United States over Russian Federation now.
Peskov echoed that sentiment in the CNN interview, but urged the United States to do more to kickstart the dialogue.
Several congressional committees, as well as U.S. intelligence agencies and the Federal Bureau of Investigation, continue to investigate how and to what degree Moscow may have helped Trump score his completely unexpected victory over Democratic presidential nominee Hillary Clinton.
"He was talking about bilateral relations", Peskov said. He was talking about what is going on in the United States so we have a better understanding in Moscow, ' Peskov continued. "This is what is being performed by every ambassador of Russian Federation overseas, every ambassador of the United States overseas, including in Moscow".
Peskov also said Putin never voiced support for then-presidential candidate Trump.
Earlier in March, Senator Richard Blumenthal said that appointing a special prosecutor for the probe into alleged Russian interference in the U.S. election and ties to the President Donald Trump campaign was critical for the integrity of the Justice Department.
"If you look at some people connected with Hillary Clinton during her campaign, you would probably see that he had lots of meetings of that kind", he said.
Peskov, however, decried the "hysteria" of the USA political debate, and said it is hampering efforts to establish warmer relations.
The spokesman for Russian President Vladimir Putin on Sunday confirmed a Kremlin envoy met with members of the Trump campaign during the election to discuss "bilateral relations" - but he also contacted people linked to the Clinton camp.
"We are losing potential by blaming everything on Earth on each other".