The Moscow news conference came after Mr Putin met Mr Tillerson for the first time since Mr Trump took office.
According to the report, the official underlined that the US did not know in advance that the attack was going to take place.
But despite the growing rifts, some general compromises were discussed.
Russian Foreign Minister Sergei Lavrov said Putin might agree to resume an information-sharing "deconfliction" network with the United States on the countries' warplane flights in Syria.
Syrian President Bashar Assad said an alleged chemical attack blamed on his regime last week in Idlib province was "100 percent fabrication" used to justify a USA airstrike, news agency AFP reported on Thursday.
Tillerson conceded the two world powers had "sharp differences" that have obstructed cooperation but voiced optimism they could make progress.
"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", he said.
Lavrov had a ready retort: "This is obviously the subject where our views differ".
"Where is the proof that Syrian troops used chemical weapons?"
He said that Russian Federation hopes the United States can give constructive response to this suggestion. "There is a low level of trust between our two countries". One was the United States re-affirm it is committed to attacking the Nusra Front, an al-Qaida affiliated terrorist group that is also fighting to remove Assad from power in Syria.
Borini grabs late equaliser to give Sunderland hope
Substitutes:Havard Nordtveit (for Snodgrass, 81 mins) 6, Jonathan Calleri (for Ayew, 90 mins) 6. The Sunderland equaliser was a odd affair when Wahbi Khazri scored direct from a corner kick.
"People, not just in the West but across the Middle East and the world are speaking out against Assad's brutality", she said. Russia's Foreign Ministry dismissed Tillerson's remarks Wednesday. During his 2016 campaign, Trump said he would decide whether to honor the commitment to protect the Baltic republics against Russian aggression, based on whether those countries "have fulfilled their obligations to us".
The Trump administration on Tuesday said it had collected intelligence that purportedly proved Syrian forces had carried out the deadly chemical weapons attack in the northern Idlib province that led to the US missile strike.
He also backed USA calls to find a political solution and said the G7 leaders were ready to work with Russian Federation to end the violence in Syria.
But such hopes appeared optimistic as the diplomats outlined their sharply diverging views on Syria.
Russian President Vladimir Putin says relations between Moscow and Washington have deteriorated in the early months of Donald Trump's presidency.
"If they [Syria] use chemical weapons, they are going to pay a very, very stiff price", he said.
"We don't want a new Cold War", he said standing next to Trump.
U.S. Central Command said Thursday that coalition aircraft were given the wrong coordinates by their partner forces, the predominantly-Kurdish Syrian Democratic Forces, for a strike meant to target Islamic State militants south of their Tabqa stronghold. Russia's alleged meddling in the USA presidential election also hovered over the first face-to-face encounter between Putin and a Trump administration Cabinet member. The official said the likelihood is the Russians are more careful in their communications to avoid being intercepted.
The proposed resolution would have condemned the alleged attack and expressed the council's full backing to investigators from the Organization for the Prohibition of Chemical Weapons (OPCW).
CORRECTS DATE TO APRIL 12 - United Nations Security Council members show hands for a vote on a resolution condemning Syria's use of chemical weapons at U.N. headquarters on Wednesday, April 12, 2017.