While Nikki Haley, the US ambassador to the United Nations, described regime change in Syria as a USA priority and inevitable, Tillerson suggested that last week's American airstrikes in retaliation for the chemical attack hadn't really changed USA priorities toward ousting Assad.
The United States fired dozens of cruise missiles at a Syrian air base on Friday from which it said a deadly chemical weapons attack had been launched earlier in the week, escalating the USA role in Syria and drawing criticism from Assad's allies including Russian Federation and Iran.
On whether Assad should be removed from power, McMaster said: "We are not saying that we are the ones who are going to effect that change".
But for Russian Federation or any other nation, McMaster said, supporting Syria's regime isn't in its best interest, and Russian Federation ultimately decides the kind of relationship it wants with the United States.
"What we've seen is, you know, in our meetings this week, we were told of the evidence", she said. "[What] we are saying is, other countries have to ask themselves some hard questions", he said. "That's how he's taken out by his own people, with our efforts".
He said Russian Federation should also be asked how it didn't know that Syria was planning a chemical attack since it had advisers at the Syrian airfield.
But Ms Haley warned that the USA was prepared to "do more" if needed, leaving the door open for more military intervention as the global community awaits a response from President Assad and his allies.
Lawmakers from both the Democratic and Republican parties were supportive of Trump's decision to attack the Syrian air base, but some Republican senators said they were concerned about the lack of policy clarity and Tillerson's strategy of leaving Assad's fate unresolved while concentrating on Islamic State.
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"Right now, I think everyone in the world sees Russian Federation as part of the problem", McMaster said. The change in direction comes in response to reports that Assad used chemical weapons against civilians, which President Trump said had crossed a line with him. "In fact, we were prepared to do more two days ago", McMaster said.
US -backed rebels groups have long pleaded for more USA intervention and complained that Washington has only fought the Islamic State group.
Tillerson, on the other hand, stressed that the air strike had the limited aim of deterring further use of chemical weapons. "Right now I think everyone in the world sees Russian Federation as part of the problem".
British Defense Secretary Michael Fallon criticized Russia's support of Assad, describing the chemical attack as a war crime that happened "on their watch". He had little choice: Not wanting to be seen tolerating Bashar Assad's habit of gassing children, and faced with a powerful neighbour that wants more than a "tweak" in our trade relations, the prime minister got behind the Americans, though with mildly coded language and assurances that Canada wants the Syria crisis resolved through multilateralism and diplomacy.
"Here's what I think Assad's telling Trump by flying from this base: 'FU, '" Graham said.
He said Assad and the terror group are connected.
The focus of USA policy has turned to possible Russian complicity with last week's chemical attacks which the United States says were launched from Syrian aircraft based at the Shayrat air field near Homs.