Washington and Moscow were plunged into crisis when Trump ordered a missile strike against a Syrian airbase last week after a Sarin gas attack in rebel-held Idlib province killed at least 86 people.
Russia condemned the strike, calling it a violation of international law. Putin has said he believes the gas attack was actually carried out by rebels in a bid to frame the government and provide a pretext for US intervention.
But Trump turned up criticism of the Kremlin's stance on Tuesday night, saying its support for Assad was "bad for mankind."
"Frankly, Putin is backing a person that's truly an evil person," Trump told the Fox Business Network, referring to Assad. "I think it's very bad for Russia. I think it's very bad for mankind."
Of Assad, Trump added: "This is an animal."
However, he said he did not want to "get involved"in a military confrontation with Assad by sending troops into Syria.
Sean Spicer, a White House spokesman, later clarified that did not mean ruling out further missile or air strikes if Mr Assad used gas again.
Tillerson arrived in Moscow on Tuesday evening on a mission to persuade the Kremlin to drop its support for Syrian President Bashar al Assad.
Lavrov said ahead of the meeting yesterday morning that the US had carried out an unlawful attack against Syrian President Bashar Assad's forces and that Russia had lots of questions about the Trump administration's "very ambiguous"and "contradictory"ideas.
"We have seen very alarming actions recently with an unlawful attack against Syria," Lavrov said. "We consider it of utmost importance to prevent the risks of replay of similar action in the future."
Tillerson, the first Trump Cabinet official to visit Russia, conceded the US and Russia had "sharp differences" that have been obstructing co-operation between the two powers. He said he was seeking to "better understand"the differences so that they could be narrowed.
"We both have agreed our lines of communication shall always remain open," Tillerson said.
Tillerson reiterated the US view that Bashar al-Assad was responsible for the April 4 gas attack.
"I think the perspective from the US is supported by the facts we have, which are conclusive: the recent attack was executed and planned by Syrian regime forces. I think the characterisation is one that Assad has brought upon himself [the US strikes]" he said. He also said the US believed the Assad regime had used chlorine gas on civilian populations on "more than 50 occasions".
Lavrov said that Russia required a proper UN investigation to establish the truth of the attack.
"I am not trying to level accusations at anyone or exonerate anyone. I am just saying that we should have an investigation to establish what happened on April 4," he said.
