There are several questions Mr. O’Rourke is considering aloud, a person close to him said: Could he build a full-scale national campaign without losing the down-home feel that powered his Senate bid, when fans tracked his 254-county tour of Texas (down to the four-hour drives and late-night burger runs) on a near-constant video feed? Would a hope-and-change chorus find an audience in a primary with other Democrats — and without an easy Republican foil like Mr. Cruz?

Yet Mr. O’Rourke also plainly recognizes two truths about politics in the age of Mr. Trump: Traditional qualifications to lead the country do not necessarily matter much, particularly if a candidate can channel the kind of enthusiasm that Mr. O’Rourke earned in a news media environment that prizes viral moments. And politicians rarely get shinier over time; his best shot at the White House, if recent history is a guide, may be this one.

“Democrats fall in love,” said Gene Martin, a local Democratic chairman in New Hampshire, describing a “pause” in 2020 staffing activity in the state while Mr. O’Rourke makes up his mind. “He would get a king’s welcome.”

At the same time, Mr. O’Rourke’s flirtation is dividing some liberals who wonder if a white man with his résumé and biography is the best fit for this moment, just after the party recaptured the House, in large measure, on the strength of female and nonwhite candidates.

“What is it with a party that gets excited about a guy who loses but tries to undercut somebody who wins?” said Rahm Emanuel, the mayor of Chicago, describing the opposition among some Democrats to Nancy Pelosi’s continued leadership in the House, despite her presiding over a midterm wave. “Our party is emotional.”

Other Democrats eyeing 2020 have said little about Mr. O’Rourke, as is customary with a possible rival in a race that has not begun. But several of their advisers said Mr. O’Rourke’s popularity with donors and activists was undeniable and could create early momentum in key states, his prospects buoyed by a Trump-level command of social media and a talent for making the generically progressive sound inspirational.