LUCCA, Italy — Secretary of State Rex W. Tillerson said on Tuesday that President Bashar al-Assad’s reign in Syria was “coming to an end,” and he warned that Russia was at risk of becoming irrelevant in the Middle East by continuing to support him.

His remarks, five days after the United States carried out missile strikes in retaliation for a chemical attack for which Washington and its allies blamed Mr. Assad, illustrated the extent to which the Trump administration has, in just one week, substantially rethought its approach to Syria’s future.

Before the April 4 chemical attack, the administration appeared resigned to letting Mr. Assad’s government, backed by Russia and Iran, continue gaining the upper hand in a six-year-long civil war that has claimed at least 400,000 lives. Even after the attack, Sean Spicer, the White House press secretary, said the administration would look “rather silly not acknowledging the political realities” of Mr. Assad’s grip on power.

But then, late last week, Mr. Trump declared that efforts to change Mr. Assad’s behavior had “failed very dramatically.” Mr. Tillerson said that Mr. Assad could no longer remain in office, and that “steps are underway” for an international effort to remove him. On Tuesday, Mr. Tillerson seemed close to embracing the very policy the Obama administration had decided on: that Mr. Assad would eventually have to cede power, though the timeline remains unclear.