Earlier in an interview broadcast on Wednesday before the meeting with Tillerson, Putin said that mutual trust between Russian Federation and the United States, especially on the military level, had eroded in the first few months of Trump's presidency.
Wayne White, former deputy director of the State Department's Middle East intelligence office, told Xinhua that the idea of Washington and Moscow reaching a consensus is highly unlikely.
The White House wants to see Russian Federation drop its support for Assad, whom Trump labelled a "butcher".
During his campaign, then President-elect Donald Trump pledged that he would have a much better relationship with Russian Federation than President Barack Obama had.
Western leaders backed the USA action, saying Assad had brought it on himself.
Describing the Syrian leader as a "butcher" he said he had "absolutely no doubt" the U.S. had done the "right thing". "It may be just the opposite".
US Secretary of State Rex Tillerson's visit to Moscow comes a few days after his country attacked from the sea a Syrian airbase in retaliation for the use of a chemical weapon on a remote town in the Idlib province, blamed on the Assad regime.
Using its veto power, Russian Federation not only blocked the United Nations from investigating a recent chemical attack in Syria, but it kept the U.N. from even condemning it. Trump said it was "certainly possible" though "probably unlikely".
The news conference came after Putin met Tillerson for the first time since Trump took office.
Earlier in an interview broadcast on Wednesday before the meeting with Tillerson, Putin said mutual trust between Russian Federation and the USA, especially on the military level, had eroded in the first few months of Trump's presidency.
Syria's government has denied responsibility for the gas attack in a rebel-held area of northern Syria that killed at least 87 people, many of them children. Lavrov said United States allegations that Syria carried out the chemical attack "do not correspond to reality".
The Syrian regime and its backers, including Russian Federation, have denied President Bashar al Assad was responsible. Lavrov then described a list of USA efforts to achieve "regime change" around the world, from Serbia to Iraq to Libya. Mr Tillerson said there is a "low level of trust" between the two countries, and many issues on which they disagreed.
It was an ominous start to Tillerson's visit the first to Russian Federation by a Trump Cabinet official.
US Secretary of State Rex Tillerson was more equivocal about Assad's future on Sunday.
Tillerson called for Assad to eventually relinquish power.
"It is clear to us the reign of the Assad family is coming to an end", he told reporters in Italy.
Beyond Syria, Russia's alleged meddling in the USA presidential election also hovered over what was the first face-to-face encounter between Putin and any Trump administration Cabinet member.
Russian Foreign Minister Sergey Lavrov blasted USA claims that it has "irrefutable evidence" of election interference. "With all the existing problems, both objective and artificially created ones, we still have quite a few prospects for joint work". "I do not know who saw them".