GREEN BAY, Wis. — In June 2012, the morning after Scott Walker became the first governor ever to survive a recall election, the talk of higher office began in earnest.

Some conservatives said his victory instantly placed him in the mix of potential Republican candidates for president in 2016. Then came a memoir, and then a trip with other potential candidates to a meeting widely understood to be an audition before Sheldon G. Adelson, the casino billionaire and top Republican donor.

But that was then.

Now Mr. Walker, 46, finds himself in a political corner, locked in a rough fight to hold on to his job. But as he battles Mary Burke — a Democrat who was once the state’s commerce secretary, appointed by former Gov. Jim Doyle, but barely known statewide until this campaign — Mr. Walker’s day job is not all that is at stake. His currency as a presidential contender will surely vanish if he cannot win a second term as the governor of Wisconsin.