Tennessee Democrats faced near extinction under President Barack Obama, losing House seats they had controlled for generations and swapping near-majorities in both chambers of the state legislature for the barest of minorities.

Should Mr. Bredesen become the party’s standard-bearer, however, he will test whether the state has undergone an enduring realignment or whether a well-financed and well-known Democrat can win when there is a more favorable climate and a Republican president. (It is often easier for Southern Democrats to prevail when they cannot be linked to a liberal boogeyman in the White House.)

The race will also determine if Tennessee’s tradition of electing political moderates, whether they be Democrats or Republicans, was a reflection of a less polarized time or still holds as a lasting feature of a varied state that spans two time zones and absorbs Appalachia, the transient-rich New South and more agrarian Old South.

Even as it has shifted right in the same fashion of its Southern neighbors, Tennessee has continued to elect Republican pragmatists such as Mr. Corker, Senator Lamar Alexander and Gov. Bill Haslam, who briefly considered a Senate bid.

Mr. Bredesen, himself a New York transplant, will probably face one of two conservatives: Representative Marsha Blackburn, who lives in suburban Nashville and serves a district that stretches to Memphis, or former Representative Stephen Fincher, who is from rural western Tennessee. First, though, the former governor must grapple with a primary against James Mackler, a lawyer and Army veteran. On Thursday, Tennessee Republicans voiced deep skepticism that any Democrat could win statewide there.

“The Tennessee Phil Bredesen governed is a much different Tennessee than what will welcome candidate Bredesen in 2018,” said Mark Braden, a Republican strategist in the state, which Mr. Trump carried by 26 points.

But Mr. Bredesen’s advisers believe that with Mr. Obama gone from the scene, the Democrats and independents who cast Republican ballots over the last decade will again be in play when presented with a consensus-oriented moderate.