Trump was at a joint press conference on Wednesday with Nato Secretary General Jens Stoltenberg.
After the talks, Tillerson warned "there is a low level of trust between our two countries". He said "it would be a fantastic thing" if the two nations got along better but cautioned that "it may be just the opposite".
Haley, prior to the chemical attack, had said, "Our priority is no longer to sit there and focus on getting Assad out". Both the Syrian government and opposition have asked for an independent investigation Safronkov said, "whereas the OPCW is doing nothing, for reasons unknown".
On April 4, a reported toxic gas attack hit the rebel-held town of Khan Sheikhoun in Syria's Idlib, killing over 80 people and wounding scores of others.
Trump had called the military alliance, which is a bulwark against Russian Federation, "obsolete" while campaigning for the presidency.
Putin and Trump are yet to meet face to face to discuss the tensions between Russian Federation and the US.
Putin met with Tillerson in Moscow on Wednesday. Though President Donald Trump has spoken with Putin by phone, the meeting was believed to be the first between Putin and a top member of Trump's administration since the new American president took office in January.
"When you look at what happened with this chemical attack, we knew that we could not stand passive on this", he said. "I do not know who saw them".
"The Kremlin argues Assad gave up his chemical weapons stores under a 2013 agreement Moscow brokered with the Obama administration; Tillerson has accused Russian Federation of "incompetence" in failing to enforce that deal", Charles Maynes reports for NPR from Moscow. Tillerson has a meeting set later Wednesday in Moscow with Foreign Minister Sergey Lavrov.
Lavrov said global experts should be sent to the place where the chemicals were used.
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Russia's top foreign minister said Wednesday that his country will consider reopening a phone line with the United States used to avoid aircraft collisions over Syria only if the US agrees to stop attacking Syrian government targets.
But such hopes appeared optimistic as the diplomats outlined their sharply diverging views on Syria. One was the United States re-affirm it is committed to attacking the Nusra Front, an al-Qaida affiliated terrorist group that is also fighting to remove Assad from power in Syria.
He added that the reign of the Assad family is "coming to an end, and they have brought this on themselves".
"It reminds me of the events in 2003 when US envoys to the Security Council were demonstrating what they said were chemical weapons found in Iraq", Putin said. And he suggested that possible war crimes charges could be levied against the Syrian leader. "We may be at an all-time low in terms of relationship with Russian Federation".
Steeped in geopolitical intrigue, the meeting between Putin and Tillerson wasn't formally confirmed until the last minute, following days of speculation about whether the Russian would refuse to grant the former oil executive an audience.
On the eve of the talks, far from trying to calm tempers, both sides escalated their rhetoric as the United States tried to prise Moscow and Damascus apart.
The Trump administration says it is spending a disproportionate share on defense compared with its 27 partners, and that it expects action by the time Trump meets with other alliance leaders on May 25.
US President Donald Trump has described relations with Russian Federation to be at an "all-time low" following the US missile strike on a Syrian airfield last week.
"We do think it's important that Assad's departure is done in an orderly way", he continued.