At least 40 people were killed and nearly 500 treated for symptoms arising from the poisonous gas attack.
Although Trump said Monday that he would make a "major decision" in 24 to 48 hours, the timing of a USA -led strike could be delayed by Syria's announcement Tuesday that it will allow worldwide chemical weapons inspectors into the country to investigate Saturday's attack, which left about 50 people dead in the rebel-held town of Duma, as well as by the need for the Pentagon and US allies to get sufficient military forces in place.
If Trump approves a more punishing campaign, USA commanders are likely to insist on taking out Syria's air defense systems, which reportedly include sophisticated Russian-supplied S-400 ground-to-air missile batteries.
As president-elect, Trump held that line, telling supporters his administration would "stop racing to topple foreign regimes that we know nothing about, that we shouldn't be involved with".
President Vladimir Putin's spokesman Dmitry Peskov said in televised remarks that "we don't participate in Twitter diplomacy". There was also no indication that Syrian government forces had entered Douma on Thursday, the Associated Press reported.More news: NZ stops offshore oil and gas exploration to tackle climate change
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Earlier, a spokeswoman for the Russian Foreign Ministry responded to Trump on Facebook, saying missiles would destroy "all evidence" of the attack. On Wednesday, Maria Zakharova, the Russian foreign ministry spokeswoman, said that if the US missiles were so smart then they should hit "terrorists" and not government targets. In April 2017, as he contemplated a strike in Syria, Trump said, "One of the things I think you've noticed about me is: Militarily, I don't like to say where I'm going and what I'm doing".
"If it's Russian Federation, if it's Syria, if it's Iran, if it's all of them together, we'll figure it out and we'll know the answers quite soon", he pledged Monday. Get ready Russian Federation, because they will be coming, nice and new and "smart!", Mr Trump tweeted.
Trump's tweet also addressed United States action against ISIS, who were until recently one of the biggest threats in Syria.
Mrs May had agreed with Mr Trump and French President Emmanuel Macron that a chemical weapons attack, believed to have been carried out by the Assad regime on Saturday and which killed 43 people, could not go unchallenged. The Syrian government and its ally Russian Federation have denied that such an attack ever happened.
Both Trump and his defense secretary Jim Mattis abruptly canceled upcoming travel plans Tuesday, as the USS Donald Cook - a guided-missile destroyer - moved to within striking range of Syria.
US officials tell The Associated Press that the allies have weighed launching a military strike as early as the end of this week. "The President has a number of options at his disposal and all of those options remain on the table".
May recalled ministers from their Easter holiday to debate action over what she has cast as a barbaric poison gas attack by Syrian government forces on civilians in the formerly rebel-controlled town of Douma, just east of the capital Damascus. Syria denies the accusation.