Bredesen himself is a superdelegate and undecided in the presidential race. The national convention will have almost 800 superdelegates -- elected and party officials -- whose votes for a presidential nominee are not bound by the results of any primary or caucus.
Democratic National Committee officials say that the idea of a superdelegate primary has been floated by other superdelegates as well in the past few weeks but that the superdelegates have mixed feelings about the idea. Although it would help avoid a delegate fight at the convention, some argue the convention is a superdelegate primary, and the rules say they don't have to make a decision until then.
"I'm very concerned about what would happen if you got just sort of a brutal summer of two people battling it out," Bredesen said in an interview with The Associated Press on Wednesday.
Bredesen, who is also policy chairman of the Democratic Governors Association, is proposing a two-day superdelegate primary for June after the last voter primary.
"There's going to be wounds -- there are wounds already -- and we need to be healing them in June, July and August, and not waiting until two months before the election," he said.
Obama leads Clinton among delegates whose vote was determined by primaries or caucuses, 1,404 to 1,249. But neither is on track to win enough delegates in primaries and caucuses to clinch the nomination, so the superdelegates could decide the outcome of the contest.
Clinton leads among superdelegates who have announced a choice, 250-213.
Bredesen, who first proposed the idea in an opinion piece in Wednesday's editions of The New York Times, said he has spoken about it with Democratic National Committee Chairman Howard Dean.
"He was very open and complementary and urged me to go ahead and continue making the case for this," Bredesen said.
Bredesen acknowledged that the superdelegate primary doesn't have a chance of becoming a reality if it isn't run by the DNC.
The governor said he has also spoken to Clinton, who "certainly wasn't repelled by the idea," he said. He planned to speak with Obama later Wednesday.
Bredesen was also planning to make his case to Al Gore, the party's 2000 nominee and one of the most prominent of the remaining uncommitted superdelegates.
About 40 percent of the superdelegates have not declared their choice, including 10 Democratic governors. Bredesen said some sort of intervention is needed to force the issue.
"The best way to do it is to put some enormous pressure -- moral and otherwise -- on all the superdelegates, myself included," he said. "It's time to fish or cut bait here."
Bredesen called for a "business-like" superdelegate gathering.
"There would be a final opportunity for the candidates to make their arguments to these delegates, and then one transparent vote," he said in the op-ed piece. "This is our electoral process at work in a way the founders would be proud of."