The call came Thursday, after Putin used his annual news conference to praise the US financial markets' rise and other "quite serious achievements" under Trump, whom he said he was on first-name terms with.
Mr Tillerson also called on China and Russian Federation to increase pressure on North Korea by going beyond the implementation of United Nations sanctions. "That needs to be done by the voters, the American people", Mr Putin said.
Russian President Vladimir Putin told reporters at an end-of-the-year press conference in Moscow on Thursday that the sweeping investigation into possible collusion between his government and the Trump campaign is a "delirium" cooked up by Trump's critics.
The White House added that both "discussed working together to resolve the very risky situation in North Korea".
The two men had also talked about improving dialogue between Russian Federation and the United States, Peskov said.
He also accused those investigating potential collusion between Russian Federation and the 2016 Trump presidential campaign of damaging the U.S. political situation, "incapacitating the president and showing a lack of respect to voters who cast their ballots for him". But Russia is inextricably linked to any discussions regarding North Korea, so Trump also doesn't have much of a choice.
Washington and Moscow had many common interests, he said, citing the Middle East, North Korea, global terrorism, environmental problems and the proliferation of weapons of mass destruction. "The pressure campaign must and will continue until denuclearization is achieved".
The Security Council has ratcheted up sanctions on North Korea over its weapons programs since 2006.