On the Republican side, South Dakota Sen. John Thune and South Carolina Sen. Lindsay Graham rejected the idea, while conservative Kansas Sen. Sam Brownback expressed hesitation because of concerns over voter perception, the congressional daily publication The Hill said Tuesday referring to an April survey.
For Democrats, Virginia Sen. Jim Webb said he wouldn't consider serving as a running mate. Former governor and Indiana Sen. Evan Bayh was a bit more elusive.
"I suspect that's not the sort of thing you say no to," he said.
Conventional wisdom among political observers say both party candidates would likely pick a governor, or former governor, as a running mate.
Some joked about the low-profile position as the No. 2 person in the White House.
"Of course," Sen. Bob Bennett, R-Utah, said. "Big house, big car, not much to do. Why not?"
"Absolutely," said Sen. Barbara Mikulski, D-Md. "I think I would be great. First of all, I know how to behave at weddings and funerals."
"Once is enough," Lieberman said. "I already have the T-shirt and I'm proud of it."
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