Yet there is no evidence of any growing public angst about socialism sweeping the United States. As a political philosophy and organizing tool, it took modest root in the country in the late 19th and early 20th centuries, but never gained widespread appeal. Eugene V. Debs, a labor leader from Terre Haute, Ind., was a five-time candidate for president, never to great effect, peaking at 6 percent of the vote in 1912.

“You really have not had a self-consciously socialist movement of any size and influence since the 1930s,” said Michael Kazin, a professor of history at Georgetown University and the author of a history of the American left. “Clearly this is an attempt to portray Democrats as too radical for Americans and to connect them to Venezuela, which is of course a clever thing do since Venezuela is falling apart under an ostensibly socialist government.”

But the supposed threat of creeping socialism — and the dangers posed by someone like Ms. Ocasio-Cortez — have become favorite talking points for conservatives like the TV personality Sean Hannity of Fox News, who tells his viewers that far-left socialism had taken over the Democratic Party. Mr. Trump is now firmly aligned with that view.

“Most Americans are obviously not up on the distinctions between democratic socialists and communists,” Mr. Kazin said. “He, like other conservatives who had talked about the so-called Red Menace over the years, is trying to confuse the two things in people’s minds.”

But that is not how conservatives look at the way Mr. Trump seems ready to portray Democrats.

“From a political standpoint, he is defending free enterprise, free markets and freedom,” said Greg Mueller, a conservative strategist. “They want to take the country toward socialism and their party is divided on that and there is a major fight in their party over whether to be a socialist party.

“This is a great debate for Trump to define in 2019 and the 2020 campaign,” he added.

Republicans, with limited success, tried at times to label President Barack Obama a socialist, particularly for his call for higher taxes on the wealthy, Mr. Kazin said, noting that this effort coincided with a shift in public opinion where Americans viewed socialism more favorably.

But a Gallup poll in August showed that Democrats had a more positive view of socialism than they do of capitalism, 57 percent to 47 percent. Their view has been relatively stable since 2010, but attitudes toward capitalism have become more negative, coinciding with the financial crisis that fueled animus toward the large banks and investment firms blamed for the economic devastation.