The president tweeted: “Get ready Russia, because they will be coming, nice and new and <<smart>>!”
Senior Russian figures have threatened to meet any US strikes with a response.
The Syrian government denies mounting a chemical attack on Douma.
In one his tweets on April 11, President Trump called Syrian President Bashar al-Assad a “gas killing animal”.
In another, President Trump painted a dark picture of US-Russia relations but said it did not have to be that way.
He tweeted: “Our relationship with Russia is worse now than it has ever been, and that includes the Cold War. There is no reason for this. Russia needs us to help with their economy, something that would be very easy to do, and we need all nations to work together. Stop the arms race?”
Meanwhile, the US, UK and France have agreed to work together and are believed to be preparing for a military strike in response to the alleged chemical attack at the weekend.
Syrian opposition activists and rescuers say government aircraft dropped bombs filled with toxic chemicals on Douma.
According to the Syrian-American Medical Society (SAMS), which operates in rebel-held areas, and local aid workers, more than 500 people had been treated for symptoms “indicative of exposure to a chemical agent”.
On April 11, the UN’s World Health Organization demanded access to verify reports from its partners, which include SAMS, that 70 people had died – including 43 who showed “symptoms consistent with exposure to highly toxic chemicals”.
Meanwhile, a team from the Organization for the Prohibition of Chemical Weapons (OPCW) is due to deploy to Syria “shortly” to determine whether banned weapons were used.
The town of Douma, the last major rebel stronghold near the capital Damascus, was under renewed assault from Syrian and Russian forces last week.
Rebels have now been evacuating Douma under an agreement involving the Russian military.
Russia said it would deploy military police to Douma on April 12 and that the situation there had stabilized.
Several senior Russian figures have warned of a Russian response to a US attack, with Alexander Zasypkin, Moscow’s ambassador to Lebanon, repeating on April 11 a warning by the head of the military that missiles would be shot down and their launch sites targeted if they threatened the lives of Russian personnel.
Also on April 11, Russian foreign ministry spokeswoman Maria Zakharova asked whether the aim of Western strikes might be “to quickly remove the traces of the provocation… [so] international inspectors will have nothing to look for in terms of evidence”.
Addressing new ambassadors in Moscow, President Vladimir Putin said the world was becoming more chaotic. He said he hoped common sense would prevail and that the situation would stabilize.
President Putin said Russia would “keep all its international obligations in full”.
On April 10, President Trump cancelled his first official trip to Latin America so he could focus on Syria.
On April 11, Defense Secretary James Mattis said the US was still assessing the chemical attack and that the US military stood ready “to provide military options if they are appropriate as the president determines”.
France’s President Emmanuel Macron said any strikes would “not target allies of the [Syrian] regime or attack anyone, but rather attack the regime’s chemical capabilities”.
However, The Times newspaper reports that the UK’s PM Theresa May has urged President Trump to provide more evidence of the suspected chemical attack.
A US Navy guided-missile destroyer, the USS Donald Cook, is in the Mediterranean Sea.
On April 10, the UN Security Council failed to approve moves to set up an inquiry into the alleged attack on Douma.
As permanent members of the council, Russia and the US vetoed each other’s proposals to set up independent investigations.
The US-drafted resolution would have allowed investigators to apportion blame for the suspected attack, while Russia’s version would have left that to the Security Council.
The OPCW’s fact-finding mission will not seek to establish who was responsible for the attack.
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