Senior Russian figures had threatened to meet any US strikes with a response.
Mr Trump had promised a "forceful" reply to the suspected attack.
President Bashar al-Assad's government, which receives military backing from Russia, denies being behind any chemical attack.
In his tweet, Mr Trump called Mr Assad a "gas killing animal".
On Saturday, Syrian opposition activists, rescue workers and medics said the rebel-held town of Douma in the Eastern Ghouta region had been attacked by government forces using bombs filled with toxic chemicals.
The Syrian-American Medical Society said more than 500 people had been found with symptoms "indicative of exposure to a chemical agent", and on Wednesday the World Health Organization demanded access to the area to verify reports from its partners that 70 people had died.
After the attack, Mr Trump and his Defence Secretary, James Mattis, cancelled travel plans.
The US and Western allies said they have agreed to work together.
But several senior Russian figures warned of a Russian response to a US attack, with Alexander Zasypkin, Moscow's ambassador to Lebanon, repeating on Wednesday a warning by the head of the military that missiles would be shot down and their launch sites targeted.
What happens next?
The US and allies - France and Britain - are believed to be preparing for a military strike.
US President Donald Trump cancelled his first official trip to Latin America so he could focus on Syria.
That decision suggests the US response may involve a larger military operation than a limited strike.
French President Emmanuel Macron said any strikes would target Syrian government chemical facilities.
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