Getting selected to run for vice president of the United States requires a most unpolitical skill: keeping your mouth shut.

With time running out for Sens. Barack Obama (D-Ill.) and John McCain (R-Ariz.) to name their running mates before the political conventions, senators are doing their best to honor that cardinal rule. With a few notable exceptions, senators believed to be on the candidates’ short lists are clamming up when asked about their interest in the job.

Like many, Dodd said the No. 1 rule for a potential vice president is the ability to “keep your mouth shut.”

Republican Sen. John Thune of South Dakota, whose youth and conservative credentials have propelled him into speculation as a potential running mate for McCain, also downplayed his interest. Yet like Dodd, he won’t rule it out.

“I don’t have any designs on the job, I haven’t campaigned for it, and I am very, very happy with the job I have,” he said. “You don’t want to foreclose options, but I’ve also made it abundantly clear that it’s not something that I want to do and not something I aspire to do. And I’m not doing anything to give people any reason to believe I am.”

Democratic consultant Bob Shrum, who worked on the presidential campaigns of Richard Gephardt, Edward Kennedy, Jimmy Carter, Michael Dukakis, John Kerry and Al Gore, said senators like Dodd and Thune are following a rule with a reason.

“The one rule is to not look like you’re campaigning for it,” Shrum said. “You don’t want to [make] the presidential nominee look like he’s yielding to a campaign.”

Some senators and potential picks have already taken themselves out of the game. Democrat Jim Webb of Virginia issued a statement earlier this month that said he won’t be a candidate, and Louisiana Gov. Bobby Jindal (R) said the same about joining a McCain ticket. Jindal even timed his statement just ahead of a private visit by McCain to Baton Rouge.

“I have said in every private and public conversation, I’ve got the job that I want,” Jindal told Fox News. “And I’ll say again on air: I’m not going to be the vice presidential nominee or vice president.”

Shrum says such statements by Webb and Jindal exemplify another time-honored tradition: Sometimes it’s better to rule yourself out before you get ruled out.

“By putting out a public statement, you make it clear — and it doesn’t look like you went through the process and didn’t make it,” Shrum said. “Almost everybody says no at first anyway. They say, ‘I really love my present job,’ or ‘I have no interest in it.’ It’s almost ritualistic, and it’s self-protection. It’s another way you don’t look like you’re campaigning for it.”

One problem some senators have: They’ve never been in the position before of being a vice presidential possibility. Earlier this year, for example, GOP Sen. Kay Bailey Hutchison of Texas sought to end speculation that she would join the Republican ticket by consistently stating in interviews that she didn’t want the job.

Hutchison said she was simply following her instinct, not any unspoken rule.

“It’s a very individual thing, and people don’t quite know what to say. If you’re not pursuing it, how do you not pursue it?” Hutchison said. “In my case, it just wasn’t something I saw for my future.”

Senate Republican Conference Chairman Lamar Alexander of Tennessee, a former governor and presidential candidate, recalled being vetted by George W. Bush before the 2000 campaign. Alexander, who had served Bush’s father as education secretary, agreed with Shrum that the first rule for a running mate is discretion.

“The worst thing you can do is publicly seek it,” Alexander said. “Probably the best thing to do is to keep your mouth shut and if you get invited to go through the vetting process, you go through it and then you just be yourself.”

Democratic Sen. Claire McCaskill of Missouri said keeping a low profile represents an important shift in attitude for potential vice presidential candidates. McCaskill has campaigned with Obama, and her gender and Midwestern appeal have made her an attractive possibility for the Democratic ticket.

“What most people do is they go through an exercise in their head of thinking — rather than what’s best for them — they think about what’s good for the campaign,” McCaskill said. “And anyone who has much sense understands what’s best for the campaign is for Barack Obama to be left alone and make a wise choice based on his good judgment.”

Both Thune and McCaskill told “Fox News Sunday” that they were unaware of any vetting by the McCain or Obama campaigns — but neither ruled it out.

“So you’d say yes?” asked host Chris Wallace.

“Well, you know, I would like to meet somebody who wouldn’t,” said McCaskill.