Although the G-7 foreign ministers agreed on bringing a swift end to Syria's conflict, Russia's cooperation is necessary to solve the situation. USA officials in Washington have also raised that prospect.
With a growing gulf between the USA and Assad's main ally Russian Federation, the G7 ministers threw their support behind Tillerson as he headed for vital talks on the Syrian conflict in Moscow. "I welcome this renewed harmony".
US Treasury Secretary Steven Mnuchin said last week that the USA would soon impose additional sanctions on Assad, as part of a military, diplomatic and financial response to a chemical attack blamed on the regime.
The White House separately suggested that barrel bombs dropped by Assad's forces could be one of the military actions in Syria that could prompt a United States response - if chemicals were involved.
Mr Lavrov also said there were "confusing and sometimes openly contradictory ideas" from the Trump administration on the US-Russia relationship and global issues. Moscow is willing to discuss all matters of mutual concern raised by Tillerson and is ready "for any course of events", the ministry noted.
And - Russian President Vladimir Putin has compared recent accusations of a chemical attack leveled at the Syrian government how the United States justified its intervention in Iraq in 2003. The senior official said it still wasn't clear who was flying the jet that bombed the hospital.
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The United States launched a missile strike on a Syrian airfield on Friday in its first military action against Assad in retaliation for the attack.
Moscow pledged to the USA in 2013 that it would work to rid Syria of its chemical weapons arsenal.
US Treasury Secretary Steven Mnuchin said last week that the US would soon impose additional sanctions on Syria, as part of a military, diplomatic and financial response to a chemical attack blamed on the regime.
U.S. Secretary of State Rex Tillerson steps off his plane at Pisa Military Airport in Pisa, Italy, on Sunday, April 9, 2017.
While the claim from the governor of Syria's Homs province may or may not have been true, what is true is that Secretary of Defense James Mattis said the singular attack destroyed 20 percent of Syria's operational air force.
"We have reports from multiple sources that false flags like this one - and I can not call it otherwise - are being prepared in other parts of Syria, including the southern suburbs of Damascus".
Following on from the Tillerson visit, Moscow has announced Lavrov will host three-way talks with his counterparts from Syria and Iran later this week.