But the American ambassador to the United Nations, Nikki R. Haley, suggested that such a process was doomed as long as Mr. Assad was in power. “We know there’s not any sort of option where a political solution is going to happen with Assad at the head of the regime,” she said on CNN. “If you look at his actions, if you look at the situation, it’s going to be hard to see a government that’s peaceful and stable with Assad.”

That statement stood in contrast not only to Mr. Tillerson’s comments but also to Ms. Haley’s own remarks a week ago — before Mr. Assad carried out his latest chemical weapons attack on civilians — in which she insisted that his departure from office was not a diplomatic priority for the United States.

Still, the overall tone of suspicion and condemnation of Russia’s actions in Syria indicated that Mr. Trump’s top national security advisers were nudging him back to a more traditional Russia policy. During his days as the chief executive of Exxon Mobil, Mr. Tillerson received a friendship award from Mr. Putin, and he is aware of the suspicions surrounding those ties and has gone the furthest in the administration in separating himself from the Russian leader.

The challenges have only multiplied in recent days. The Russians, angry about the attack on the air base, have threatened to cut off a communication line that the American and Russian militaries have used to notify each other about air operations in Syria. And the attack has forced Mr. Putin into a tighter relationship with Mr. Assad, perhaps tighter than the Russian leader wants.

Ms. Haley, who, like Mr. Tillerson, is new to diplomacy, has also apparently concluded that a hard line toward Russia is the safest course. The contrast between her remarks and Mr. Trump’s warm words for Mr. Putin on the campaign trail — as well as his refusal to acknowledge Russian meddling in the 2016 presidential election — has been striking.

The Trump administration’s Syria policy has been difficult to parse. Mr. Tillerson, in his first television appearances since taking office, seemed to describe two different strategic objectives: halting chemical attacks and ultimately negotiating a cease-fire. But he made it clear that he had no intention of backing a military intervention that would overthrow Mr. Assad. That suggested that as long as the dictator used conventional means to kill his own people — barrel bombs instead of sarin gas — the United States would keep its distance.

“I think what the United States and our allies want to do is to enable the Syrian people to make that determination” about Mr. Assad’s fate, Mr. Tillerson said on CBS’s “Face the Nation” — a line that was often used by his predecessor in the Obama administration, John Kerry. “You know, we’ve seen what violent regime change looks like in Libya and the kind of chaos that can be unleashed.”