Peskov also said Putin had given Tillerson his views on the situation in Syria, and how it was likely to develop.
"Right now we're not getting along with Russian Federation at all", Mr Trump said flatly during a White House news conference with Nato Secretary-General Jens Stoltenberg. -Russian relations to one of the lowest points since the Cold War.
"The longer-term status of President Assad will be decided by the Syrian people", Tillerson had said, when asked whether the US would get involved in Syria's internal politics on March 30. "So I felt we had to do something about it".
Russian Foreign Minister Sergey Lavrov blasted US claims that it has "irrefutable evidence" of election interference.
Moscow initially said Putin would not meet with Tillerson, but later relented.
U.S. Central Command said Thursday that coalition aircraft were given the wrong coordinates by their partner forces, the predominantly-Kurdish Syrian Democratic Forces, for a strike meant to target Islamic State militants south of their Tabqa stronghold.
But Trump did not seem to change in his assessment of Russian President Vladimir Putin, who Trump has spoken of admiringly in the past.
In a show of continued support for the regime, Moscow will host Syrian Foreign Minister Walid Muallem for talks with Lavrov on Thursday.
Russia's U.N. Ambassador Vladimir Safronkov told the council before the vote that during talks in Moscow, Russia asked for an independent worldwide investigation to examine the April 4 attack on Khan Sheikhoun.
Lavrov also warned Tillerson that Trump can not act in similar military fashion again in the future if he wishes to enhance diplomatic relations between Washington and Moscow, which deteriorated substantially during Barack Obama's presidency.
Chemical attack in Syria was 'fabricated': Assad
Assad said he would only allow an "impartial" external investigation involving "unbiased countries", according to AFP . Western powers blame the gas attack, which killed scores of civilians - many of them children - on Assad's forces.
"We have to see common threats" and not be guided by emotions, because "overthrowing the regime" may mean losing the battle against Islamic State, Lavrov said. But Tillerson repeated the administration's new belief that "the reign of the Assad family is coming to an end". "And they have again brought this on themselves with their conduct of the war these past few years".
"But at least we agreed to establish a dialogue on a number of important issues, including problems created by the previous administration".
US intelligence agencies have said Russian Federation hacked Democratic Party emails and leaked them in an effort to damage the campaign of Democrat Hillary Clinton and ultimately to help Trump win.
"We now understand each other better".
Mr Johnson said: "This afternoon in NY, the global community sought to make clear that any use of chemical weapons by anyone anywhere is unacceptable and that those responsible will face consequences". Ten countries voted in favor of the text, while Bolivia joined Russian Federation in voting no. "Of course, we expect the United States to respond in kind".
Western powers have lashed out at Russian Federation after it vetoed a United Nations resolution that would have condemned the suspected use of chemical weapons in northern Syria and demanded a speedy investigation. Bolivia also rejected the resolution and China did not vote. Ten nations, including the United States, voted in favor of the measure.
More than 80 people were killed in what the United States has described as a nerve gas attack that Syrian president Bashar Assad's forces undoubtedly carried out.
Trump and others have indeed threatened similar action. "Russia is engaged in a very sophisticated propaganda and disinformation campaign that is created to allow the Assad regime escape responsibility".
He said the U.S. attack on the Syrian air base "was carried out in violation of worldwide norms".
Could Syria have launched the chemical weapons attack with Russia's advance knowledge? Tillerson is a former oil executive who was awarded Russia's Order of Friendship by Putin.