"Get ready Russian Federation, because they will be coming, nice and new and 'smart!'" Mr Trump said in an early morning tweet on Wednesday.
"You shouldn't be partners with a Gas Killing Animal who kills his people and enjoys it!", he tweeted on Wednesday.
Syria referred to US threats of missile strikes as "reckless escalation", the state news agency said.
Mattis also ignored a question about whether he was concerned about telegraphing US military moves in Syria, a question appearing to refer to Trump's earlier tweet on Wednesday. The Syrian government's use of chemical weapons in suburban Douma couldn't immediately be corroborated because the area is cut off, but these attacks using outlawed weapons of war are one of Assad's signatures.
In his televised address from the White House on Friday evening, Trump said the USA was prepared to sustain economic, diplomatic and military pressure on Assad until the Syrian leader ends what Trump called a criminal pattern of killing his own people with internationally banned chemical weapons.
Explosions lit up the skies over Damascus, the Syrian capital, as Trump spoke from the White House.
At a photo-taking session during a Pentagon meeting with his Dutch counterpart, Mattis was asked by a reporter whether he had seen enough evidence to blame the Syrian government.
A French frigate, UK Royal Navy submarines laden with cruise missiles and the USS Donald Cook, an American destroyer equipped with Tomahawk land attack missiles have all moved into range of Syria's sun-bleached coast. Wednesday's statement is also notable because Trump has criticized his predecessors for publicly telegraphing U.S. military action.
Trump chastised Syria's two main allies, Russia and Iran, for their roles in supporting "murderous dictators", and noted that Russian President Vladimir Putin had guaranteed a 2013 global agreement for Assad to get rid of all of his chemical weapons.
Meanwhile, the Russian defense ministry said the Syrian regime flag was flying in Douma - the target of Saturday's attack - which it said indicated government forces have taken full control over the formerly rebel-held district of Eastern Ghouta.
Zasypkin also said that any hostilities with Washington should be avoided and Moscow was ready for negotiations. Most nations have foresworn use of chemical weapons, at least on innocent civilians.
"It is irresponsible that he would just tweet out something specific about a strike when lives are at stake, not just lives on the ground, but American lives and probably allied lives", he said.
On Thursday, Trump also pushed back against a New York Times report that he "sought to fire" special counsel Robert S. Mueller, who is leading the federal investigation into Russian meddling in the 2016 presidential election.
The Syrian government has repeatedly denied any use of banned weapons. This was the case in the last US attack against Syria, which followed a deadly sarin gas attack in the country's northwest.
Mr Assad's Barzah research and development centre in Damascus was destroyed, Mr McKenzie said.
More than 2,000 USA troops are in Syria.
The World Health Organization on Wednesday demanded "immediate" access to the victims of an alleged chemical attack in Syria, voicing indignation at the strike that caused symptoms consistent with exposure to toxic substances.