But Bredesen also indicated he's not getting involved in the election.
Wilder, 85, goes up against 51-year-old Senate Majority Leader Ron Ramsey, a Blountville Republican, in a contest to be decided by 33 senators at noon Central Standard Time.
Bredesen, who was at East Tennessee State University to help welcome the first 72 College of Pharmacy students, didn't sound passionate about the outcome of the Senate contest.
"I have this philosophical rule that you don't worry too much about things you can't control," Bredesen told reporters. "I will wait to see how it comes out. I certainly feel that of the candidates who are up there and looking at it, they are all people I can work with. ... Certainly I support the Democrat for the position, but ultimately I don't think it will make a lot of difference."
The leader of the Senate, or speaker, is also the state's lieutenant governor. The lieutenant governor stands in immediate succession to the governor's office if necessary.
The Senate speaker also has the power to make committee assignments and influence legislation - including the governor's legislative agenda. Passing a balanced budget is a yearly challenge for lawmakers, and Bredesen predicted that this year will be "a little bit of a tight year" for state revenues.
It isn't clear whether Wilder or Ramsey, who narrowly lost to Wilder in 2005, has 17 votes to win the post even though Republicans hold a slim 17-16 majority in the Senate.
Wilder has held the job since 1971. Article II, Section 3 of the Tennessee Constitution notes that the Senate speaker "shall hold his office as speaker for two years or until his successor is elected and qualified."
Tennessee Republican Party Chairman Bob Davis is backing Ramsey for the job.
"The question senators face is, do they want to move our state forward or keep Tennessee from reaching its full potential," Davis said in an e-mailed release. "I believe the people of our state are tired of settling for second best and the continued stale leadership in the General Assembly. State Senator Ron Ramsey is the kind of legislator who will provide new leadership to help move this state forward.
"A vote for Ron Ramsey is not about partisan politics. It is about educating our children, protecting our seniors, keeping taxes low, and making government more efficient. That is what this important debate is about. After 30 years, it is time our current Lt. Governor John Wilder be replaced, and any individual who thinks otherwise is playing politics."
Davis encouraged all senators, including state Sen. Mike Williams - a Maynardville Republican considered to be the swing vote in the contest - to back Ramsey.
"Anything else would be irresponsible," Davis said.
Williams' senatorial district includes Hawkins County, and that county's Republican Party has passed a resolution urging him to back Ramsey. Williams supported Wilder two years ago and then was appointed by Wilder to be Senate speaker pro tem.
"Senator Mike Williams accepted the contributions of the Republican Party, sought the nomination of the Republican Party for the state Senate, ran on that Republican Party nomination for the state Senate, and was elected to the state Senate three times now as a member of the Republican Party," according to the resolution. "Loyalty to party is both expected and requested in this circumstance."