In an interview yesterday, Putin said that if Donald Trump had meant to bring about a thaw in USA relations with Russian Federation, he has failed to see this intention through.
"Our view is that the reign of the Assad family is coming to an end and they have brought that on themselves with their conduct", the top US diplomat said.
"The secretary explained there were no subsequent targets after the missile strike, but he did not rule out any future action", State Department acting spokesman Mark Toner said in a statement.
The Kremlin had previously declined to confirm Putin would meet Tillerson, reflecting tensions over the USA strike on Syria.
"We discussed our view that Russian Federation, as their closest ally in the conflict, perhaps has the best means of helping Assad recognize this reality", Tillerson said. According to Reuters, the Russian president said his country's relationship with the U.S. has "deteriorated" under Trump, "especially on the military level".
The news conference came after Russian President Vladimir Putin met the top American diplomat for nearly two hours to see if they could rescue relations between the world's mightiest military powers.
There were only a handful of issues Tillerson and Lavrov said they broadly agreed on - that the Korean peninsula should be denuclearized, Syria should be "unified and stable" after the Islamic State is defeated, and that there should be more communication between U.S. and Russian diplomats and militaries.
Meanwhile, US Secretary of State Rex Tillerson told Russian President Vladimir Putin that the relationship between Moscow and Washington is "at a low point".
The Kremlin previously refused to say whether Putin would meet with Tillerson in Moscow.
Both Lavrov and Tillerson made mention of plans to establish a working group with representatives from the State Department and Russia's foreign ministry with the goal of improving relations between the two nations.
Meanwhile, Mr Trump has also declared North Atlantic Treaty Organisation is "no longer obsolete" - and has said he is committed to the military alliance.
Despite the dreary outlook, Trump said that he heard that Secretary of State Rex Tillerson had a positive meeting with Russian officials.
US-Russia relations are at another low after a chemical attack in Syria killed more than 80 people and wounded at least 550 others last week.
It is not the first time Mr Putin and Mr Tillerson have had talks.
"We have not seen a single fact, or even a hint of facts", he said. He said Russia's hope was that Putin's message would be passed on to US President Donald Trump. Russian Federation said the resolution had implicitly apportioned blame.