More than 20 senators say they would seriously consider an offer to be No. 2 on their party’s presidential ticket, while others claim to have little to no interest, according to a survey conducted by The Hill.

Over the past month, The Hill asked all 97 senators not running for president the same question: “If you were asked, would you accept an offer to be the vice presidential nominee?”

Senators are always in the mix of vice presidential chatter and this year is no exception, even though members of the upper chamber will be at the top of the Republican and Democratic tickets this fall.

Some of the senators who have been mentioned as vice presidential material downplayed their chances.

Republican prospects John Thune of South Dakota and Lindsey Graham of South Carolina dismissed the possibility, while conservative Sam Brownback (Kan.) expressed concern about how independent voters would react to him.

On the Democratic side of the aisle, Sen. Jim Webb (Va.) also dismissed the idea — but with a grin. Sen. Evan Bayh (Ind.), another Democrat whose name has surfaced in vice presidential speculation, was more candid: “I suspect that’s not the sort of thing you say no to.”

Some political observers expect Sen. John McCain (R-Ariz.) and Democratic front-runner Sen. Barack Obama (D-Ill.) to select a governor or former governor as their running mate. But conventional wisdom on vice presidential picks is usually wrong. George H.W. Bush’s selection of then-Sen. Dan Quayle (R-Ind.) was surprising in 1988, as was Al Gore’s 2000 pick of then-Democratic Sen. Joe Lieberman (Conn.).

Sen. Byron Dorgan (D-N.D.) noted that the Senate is a natural breeding ground for White House hopefuls: “My guess is that almost every senator looks at themselves in the mirror in the morning and sees either a future president or vice president,” he said.

While a few senators mocked the importance of being vice president, the office has become more powerful since 1992. Gore wielded a fair amount of power as Bill Clinton’s No. 2, and Dick Cheney is regarded as the most powerful vice president in history.

Yet many senators dealt with The Hill’s question with humor.

“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. And I know how to be commander in chief. I’d bring a lot of fun to the job. We would rock the Naval Observatory.”

But most senators — including Majority Leader Harry Reid (D-Nev.) and Minority Leader Mitch McConnell (R-Ky.) — rejected the idea, saying they preferred their jobs over filling Dick Cheney’s shoes.

Sens. Lamar Alexander (R-Tenn.) and John Barrasso (R-Wyo.) noted that Cheney, a Wyoming native, headed up President Bush’s vice presidential search committee in 2000 and ended up in the spot himself.

“We’ll have to see if Sen. McCain asks me to chair his selection committee. That seems to work well,” Barrasso said. “It certainly seemed to work well for the last guy from Wyoming.”

Sens. Tom Harkin (D-Iowa), Norm Coleman (R-Minn.) and David Vitter (R-La.) took the opportunity to push other candidates. Harkin likes “The Daily Show” host Jon Stewart, while Coleman promoted Minnesota Gov. Tim Pawlenty (R). Vitter favors Louisiana Gov. Bobby Jindal (R).

A couple senators indicated they have heard the question before. Sen. Kit Bond (R-Mo.) said that he was on President Gerald Ford’s short list in 1976, and Lieberman said he would defer to his colleagues.

“Once is enough,” Lieberman said. “I already have the T-shirt and I’m proud of it.”

Sens. Jim DeMint (R-S.C.) and Johnny Isakson (R-Ga.) said they’d have to check with their wives first.

“Hey, that’s an honest answer,” said Isakson.

None of the Senate’s 16 women, however, said they had to check with their families.

Sen. Larry Craig (R-Idaho), who was arrested in a Minneapolis airport bathroom sex sting and has since been ostracized by his party, responded, “I would say ‘No, Hillary.’ ”

Several GOP senators ruled themselves out because of their age, including 74-year-old Chuck Grassley (Iowa), 76-year-old Pete Domenici (N.M.) and Thad Cochran (Miss.), who is 70. If elected, McCain would be 72 on Inauguration Day.

“I’m too old to be vice president,” Grassley said. “But I am young enough to be reelected to the Senate.”

Robert Byrd (D-W.Va.), who at 90 is the chamber’s oldest member, noted that his job as president pro tempore already allows him to preside over the Senate just as Cheney does as Senate president.

“And I do not enjoy spending a lot of time at ‘undisclosed locations,’ ” Byrd said.

Senate Majority Whip Dick Durbin (D-Ill.) said he would decline a vice presidential offer since Article II, Section 1 of the U.S. Constitution forbids delegates from voting for both a president and vice president from the same state as themselves. A state’s electors can vote for a president or vice president from their state, but not both. Illinois’s delegates, therefore, would be disenfranchised by an Obama-Durbin ticket.

And Sen. Mel Martinez (R-Fla.), who was born in Cuba, noted that he couldn’t be a vice presidential nominee because the Constitution requires presidents to be natural-born U.S. citizens. Residency qualifications for a senator, under Article I, Section 3 of the Constitution, are only nine years of residency as a U.S. citizen.

Sen. Richard Shelby (R-Ala.) ruled himself out because of disinterest and said McCain would look elsewhere anyway.

“Besides, there was a famous quote about the vice presidency from Franklin Roosevelt’s vice president,” Shelby said. “You should look that up.”

Shelby was referring to John Nance Garner, who served under Roosevelt from 1933 to 1941 and described the job as “not worth a bucket of warm piss.”

Garner, who was beaten by Roosevelt in the 1940 Democratic primary, was a former Texas congressman.

The full list of the responses from all 97 senators can be accessed here .