Earlier, Russia vetoed a draft resolution at the UN Security Council presented by the US, UK and France that would have condemned last week's chemical attack in Syria and demanded that Damascus cooperate with investigators.
Tillerson insisting - Syrian leader Bashar Al- Assad's days are numbered.
Though the resolution would have provided chemical weapons investigators with access in Syria, a Russian representative to the United Nations said it didn't "serve a useful goal".
The vote coincided with U.S. Secretary of State Rex Tillerson's visit to Moscow, where he met with President Vladimir Putin and Foreign Minister Sergei Lavrov.
The pair could meet with Putin later today, state-run Tass reports, citing Kremlin spokesman Dmitry Peskov saying there's a "certain possibility" that Putin might receive Tillerson and Lavrov.
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.
Lavrov: I Told Tillerson That US Syria Strikes Contradict Anti-Terror Goal
Tillerson is considering the request, he said, "and we expect that Washington will have a constructive reaction". He said it was "unacceptable that the OPCW analyses the evidence not on location".
"We need to attempt to put an end to this steady degradation", Tillerson said.
U.S. Ambassador to the United Nations, Nikki Haley, called on Moscow to stop protecting Assad and said the United States wants to work with Russian Federation toward a political solution for Syria.
Lavrov said Moscow was ready to resume a deal with Washington to avoid incidents in Syrian airspace as the two countries lead separate bombing campaigns.
But such hopes appeared optimistic as the diplomats outlined their sharply diverging views on Syria.
"After the summit [with Chinese President Xi Jinping], it became clear that Russian Federation was not playing a constructive role in the wake of the Syrian regime's attack", the official said.
"It is clear to all of us that the reign of the Assad family is coming to an end", Tillerson declared Wednesday, openly challenging Moscow to abandon its Syrian ally. He said the recent chemical weapons attack was really the act of terrorists and rebels. And he suggested that possible war crimes charges could be levied against the Syrian leader.
Russian Foreign Minister Sergei Lavrov even calls the U.S. response "unlawful" - when the war crime of intentionally bombing hospitals has been one of Russia's signature tactics in Syria.
Final day of campaigning before crucial Turkey referendum
Under the proposed changes, Erdogan could have his term limit effectively reset and stay in power through 2029. He also appealed to voters of other parties to approve the changes so "Turkey can leap into the future".
In this image provided by the U.S. Navy, the guided-missile destroyer USS Porter (DDG 78) launches a tomahawk land attack missile in the Mediterranean Sea, Friday, April 7, 2017.
While most immediate, U.S.
While there was a "low level of trust" between the countries, Mr Tillerson pledged to increase communication between Washington and Moscow. Mr Putin had even honoured Mr Tillerson with a friendship award.
Tillerson was greeted frostily in the Russian capital as Lavrov began their meeting Wednesday by demanding to know America's "real intentions".
The agreement was suspended after the United States launched its first attack on Syria during the country's six-year conflict last week. "We believe it is of principal importance to prevent risks of a repeat of such steps in the future".
Trump and others have indeed threatened similar action. "We have already seen all of this". Both the Syrian government and opposition have asked for an independent investigation Safronkov said, "whereas the OPCW is doing nothing, for reasons unknown".
US intelligence officials speculate that Russian Federation allowed the attack to happen, although there are no known intercepts confirming the suspicion. In an interview yesterday, Putin said that if Donald Trump had meant to bring about a thaw in U.S. relations with Russian Federation, he has failed to see this intention through. "I would like to think that they didn't know, but certainly they could have". "But if I see them using gas...we have to do something".
Fleury makes late start, Penguins rush by Blue Jackets 3-1
Behind those goals, along with Marc-Andre Fleury's strong goaltending - who stopped 39 of 40 shots - the Penguins took the game. Two-and-a-half minutes later, Kessel netted a power-play goal, which gave the Penguins a 2-0 advantage over the Blue Jackets .
Putin said Wednesday that U.S. -Russia relations have "degraded" since President Trump took office.