Ms. Blackburn and her allies “know that Bredesen is popular; they know that the national brand of the Democratic Party is not,” said Kent Syler, who teaches at Middle Tennessee State and was a longtime aide to the former Democratic congressman Bart Gordon. “So they will work very hard to connect him to that national brand, and he will work very hard to avoid that.”

Mr. Bredesen, 74, jumped into the race after Mr. Corker, a two-term Republican, decided to retire. A wealthy former health care executive who served as mayor of Nashville before becoming governor, Mr. Bredesen said he concluded that he was probably the only Democrat who would have a reasonable shot at winning the seat. In a video announcing his candidacy, he reminded voters of the successes of his tenure, including reining in the soaring cost of TennCare, the state’s Medicaid program, which had been financially troubled.

His decision to run for the Senate was a coup for Democrats in Washington who have been mapping out how to flip control of the chamber in November’s elections, even as their caucus defends 26 seats, 10 in states won by Mr. Trump, while the Republicans must guard only nine. Once seen as far-fetched, winning a majority is at least mathematically possible, especially after a Democrat, Doug Jones, won a special election in Alabama in December.

Skepticism of a Democratic takeover of the Senate has come from an unlikely voice in Tennessee: Mr. Bredesen. To Republicans who suggest a vote for him is a vote to make Mr. Schumer the majority leader, he pushed back on the idea.

“I don’t think it’s possible, I’ll be honest with you,” he said, naming Florida, Missouri and North Dakota as states where Democratic incumbents could be in danger. “My expectation is that I will be in the minority in the Senate.”

Mr. Bredesen is walking a careful line when it comes to Mr. Trump, who defeated Mrs. Clinton by 26 percentage points in Tennessee. “Look, I’m not running against Donald Trump,” he says in a television commercial that is textbook Bredesen sobriety.