Tillerson reiterated the US view that President Bashar Assad of Syria, Russia's chief Middle East ally, was responsible for the chemical weapons assault in northern Syria on April 4 that left more than 80 people dead, sickened hundreds and outraged the world.
President Donald Trump painted a bleak picture of U.S. -Russia relations Wednesday, saying that the relationship between the two nations may be at an "all-time low".
Russian Federation blocked a United Nations Security Council resolution on Wednesday afternoon condemning the Assad regime's chemical weapons attack on Syrian civilians last week.
Russian Federation maintains Syrian President Bashar Assad's opposition is responsible for the chemical weapons.
"Right now we're not getting along with Russian Federation at all", Trump flatly declared at a White House press conference on Wednesday, giving stark proof the president is moving ever further from his 2016 campaign promise to establish better ties with Moscow.
"Russia is open to this, open to dialogue with the U.S. in all different areas, not only to dialogue but to joint action aimed at results in the areas where this corresponds to the interests of both countries".
He called for an "honest investigation" of the attack and said Russian Federation was "not convinced" by evidence that showed the Assad regime was culpable.
Trump indicated that it was "crazy" what was going on in the world, but that he would work to fix it.
Lavrov also said Russian Federation expects the U.S.to take part in a meeting on Afghanistan this week.
Mr Tillerson, who was personally awarded an Order of Friendship medal by Mr Putin in 2013 when he was chief executive of ExxonMobil, met Mr Putin for two hours during which time the situation in Syria and the April 4th attack were discussed.
And Assad said he could "only allow any investigation when it's impartial, when we make sure that unbiased countries will participate in this delegation in order to make sure that they won't use it for politicised purposes".
He also reaffirmed Russia's position that the Syrian government was not responsible for the chemical attack.
Secretary of state Rex Tillerson says there is a "low level of trust" between Washington and Moscow and they "cannot have this sort of relationship".
"It's very important that there is a full investigation so that everybody knows, everybody knows, so that the entire world knows how these terrible chemical attacks occurred and where they come from", said Delattre.
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