Trump himself said that Tuesday's ghastly chemical attacks changed his mind on the Assad regime.
Earlier, Tillerson spoke by phone with Russian Foreign Minister Sergey Lavrov on Saturday about the situation in Syria following Thursday's strikes on a Syrian air base, Russia's Ministry of Foreign Affairs said in a post on its official Facebook page.
As many as nine civilians, including four children were killed in the United States missile attack on a military airbase in central Syria on Friday.
US Ambassador to the UN Nikki Haley told CNN in an interview to be aired Sunday she sees no possible political solution in Syria as long as President Bashar al-Assad remains in power.
The Syria strike could be used as a model for other countries, including North Korea, he said. The Russian government, which is strongly supporting Assad, is sending warships armed with cruise missiles to Syrian naval bases as a show of strength. "There has been a swift and just response and now it is up to the (UN) Security Council, up to Russian Federation in particular, to bring this conflict to an end".
United States officials informed Russian forces ahead of the missile strikes and avoided hitting Russian personnel.
Tillerson says the Islamic State group is a threat not just to the USA but to stability in the region. The city is the target of a major offensive by US-backed and Kurdish-led Syrian forces.
So, if the new aim of the Trump administration is the removal of Assad from power - rather than just showing up Obama - this could only happen through a major confrontation with Russian Federation.
"We look forward to being together many times in the future". The foreign also suggested "the importance of an objective and unbiased investigation of all the circumstances of the tragic developments in Khan Sheikhoun and of the use of chemical weapons by militants from terrorist organizations in the past".
Trump's praise for Putin, questionable assertions about Russia's military activity in Ukraine and Syria, and insistence on a new relationship with Moscow had generated the perception that Trump wouldn't cross Putin.
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Opposition and rebel fighters, who have for years urged more direct U.S. military action in support of their uprising, hailed the strike and called for more.
The United States is leading a coalition of countries that conducts airstrikes targeting ISIS in Syria.
"At a minimum, the Russians failed to rein in the Syrian regime activity", a senior U.S. military official told reporters, speaking on condition of anonymity.
The previous day US President Donald Trump said pictures of the Khan Sheikhun victims in agony, had "an enormous impact" on him.
The strike was the first time American forces targeted a Syrian government installation in the course of the war. The base's runway was still in use.
US Treasury Secretary Steve Mnuchin said he was preparing further economic sanctions against Syria.
Russian Federation in a statement has said Moscow was suspending a Syrian air safety agreement with the United States originally drawn up to ensure that the two countries' planes did not collide.
While Trump insisted the missiles were needed for national security following a horrific chemical attack against Syrian civilians, the timing was a clear signal that the new commander in chief means business.
But he demurred on saying whether Assad had to leave power, despite the insistence of the secretary of state, Rex Tillerson, before the missile strike that diplomatic steps to oust Assad were already "under way".