It is early to speak about a shift in the Russia-US relations after a meeting between Putin and Tillerson but the talk was constructive, Peskov said.
The news conference was dominated by points of contention, chiefly Moscow's deep aversion to USA interventionism and Washington's disgust at the carnage in Syria and Ukraine.
Russian Federation and the US are now experiencing a fissure in relations after the USA conducted air strikes on a Syrian government airfield last week. "We consider it of utmost importance to prevent the risks of replay of similar action in the future". "The world's two foremost nuclear powers can not have this kind of relationship", he warned.
Peskov also said that Putin did not discuss the Ukrainian crisis in detail at a meeting with Tillerson.
The U.S. has blamed the Syrian government for launching the attack that killed more than 80, while Russian Federation has claimed that toxic agents were released from a rebel chemical arsenal hit by Syrian warplanes. Turkey has said tests showed sarin gas was used.
The tense start to Tillerson's visit illustrated the widening chasm between the former Cold War foes. Mr Tillerson said there is a "low level of trust" between the two countries, and many issues on which they disagreed. "I have absolutely no doubt we did the right thing, and it was very, very successfully done", he said. "I would like to think that they didn't know, but certainly they could have".
Meanwhile, President Trump told the Fox Business Channel that Assad was a war criminal and Russian Federation should rethink its support of him.
And Putin, who USA intelligence agencies say tried to help Trump get elected, insisted that relations with the US had only gone downhill since Trump took office in January.
And - even as Tillerson, Putin and Lavrov discussed their differences in Moscow - a fresh storm broke at the United Nations Security Council in NY.
"Two reasons why it was always likely Putin would meet Tilllerson", says Patrick Basham, director of the Democracy Institute.
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A federal task force is walking back a 2012 recommendation that men between ages 55 and 69 not be screened for prostate cancer. Why is the task force now recommending a change? USPSTF said patients are more likely to experience AEs than receive a benefit.
Also Wednesday, Lavrov said the allegations of Russian meddling in the USA presidential election haven't been proven. Russia, which backs Assad's regime, has claimed he is not responsible.
Undaunted, Trump called on United States allies to "work together to resolve the disaster" in Syria and thanked them for condemning Assad's suspected sarin attack in the town of Khan Sheikhun, which killed 87 civilians.
Writing for the Daily Sabah, Kalin evaluated the recent incidents in Syria following the regime's chemical attack in Idlib on April 4.
Russia, meanwhile, intervened in the civil war on Assad's side in 2015 and has troops on the ground, which it says are advising government forces.
United States president Donald Trump's White House victory had raised hopes in Moscow that some kind of rapprochement was possible between the former Cold War foes.
But Russia made clear it had no intention to acquiesce.
Tillerson insisting - Syrian leader Bashar Al- Assad's days are numbered.
Tillerson met Putin in the Kremlin after talking to his Russian opposite number Sergei Lavrov for around three hours.
The meeting comes as early expectations of an easy rapport between the Trump administration and Russian Federation are crashing into reality.
Only weeks ago, it appeared that Mr Trump, who praised Mr Putin throughout the United States election campaign, was poised for a potentially historic rapprochement with Russian Federation. We have stated our position on many occasions.