Almost immediately, some Democrats began to wonder if Mr. Brown possessed the secret to retaking a state that seemed to be steadily slipping from their grasp.

[Read more about the Democrats’ struggles in Ohio here.]

A week later — after insisting for years that a run for president was far off his radar — Mr. Brown has begun wondering aloud if he should have the world’s most important job after all.

Since the election, Mr. Brown said in a telephone interview this week, he and his wife, the journalist Connie Schultz, have “been overwhelmed by the number of people that have come forward and said, ‘You’ve got to run. You have the right message. You come from the right state.’”

So is he running?

“We’re thinking about it,” he said.

Rumpled and unvarnished — with a fondness for sweatshirts, less so for ties — Mr. Brown would in some ways seem uniquely positioned in a party hoping to win back the Midwestern states that flipped to Mr. Trump. Throughout his political career, he has championed populist platitudes like the “dignity of work” that have resonated with working-class voters in all corners of Ohio while also supporting liberal social causes like women’s reproductive rights and L.G.B.T.Q. rights.