Tillerson said in the news conference.
Speaking amid a backdrop of USA investigations of Russian meddling in last year's presidential election, Lavrov also said Russia confirmed its willingness to dispel problems in relations with the U.S.
The agenda for the meeting wasn't announced.
Russian Federation blocked a Western-led effort at the U.N. Security Council on Wednesday to condemn last week's deadly gas attack in Syria and push Moscow's ally President Bashar al-Assad to cooperate with worldwide inquiries into the incident.
US intelligence officials speculate that Russian Federation allowed the attack to happen, although there are no known intercepts confirming the suspicion.
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Defense Secretary Jim Mattis said Tuesday that the United States is not trying to get involved in Syria's civil war, but the US would hit Syrian government targets again if Assad used chemical weapons again.
"Right now we're not getting along with Russian Federation at all", Trump said at a White House press conference with NATO Secretary-General Jens Stoltenberg.
Looking at the resolution's supporters sitting around the horseshoe-shaped table in the Security Council, Safronkov said: "You are afraid of an impartial investigation".
"We do think it's important that Assad's departure is done in an orderly way", he continued.
The tense start to Tillerson's visit illustrated the widening chasm between the former Cold War foes.
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He said there was a "low level of trust" between the nations and "the world's two foremost nuclear powers can not have this kind of relationship".
"We want to hear concrete proof and then we will respond accordingly", Mr Lavrov said. Turkey has said tests showed sarin gas was used.
Stoltenberg said within the North Atlantic Treaty Organisation alliance, the USA military action on Syria has been met with understanding.
Could Syria have launched the chemical weapons attack with Russia's advance knowledge?
After a campaign marked by friendly comments about Russia, President Donald Trump sounded a different note on Wednesday about the current state of U.S. -Russia affairs.
The meeting Wednesday, announced by Putin spokesman Dmitry Peskov, comes hours after Tillerson faced his Russian counterpart Sergey Lavrov in a tense encounter.
"Frankly, Putin is backing a person that is truly an evil person", Trump said. "I would like to think that they didn't know, but certainly they could have".
Of Assad, Trump added: "This is an animal".
Mr Putin, speaking on Mir television, also said trust had "deteriorated" between the United States and Russian Federation under the Trump presidency.
"The level of trust at the working level, especially at the military level, has not become better but most likely has degraded", Putin said in an interview broadcast Wednesday by state television channel Mir.
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.