Let's take stock of the current crop:
Marco Rubio: Sen. Rubio is probably the politician who has built the most buzz in recent months. His endorsement of Romney recently was a help for the frontrunner and he has built a good relationship with the Massachusetts politician. Betting site Intrade gives him a 25% chance of being the VP pick.
So how does he feel?
"I'm not going to be the vice president," the Florida Republican said Wednesday at the opening of a new office - quite the strong stance.
Bob McDonnell: The Virginia governor is another leading Republican with a good rapport with Romney. He's incredibly popular and successful in his home state - a state that went blue in 2008.
McDonnell has been more receptive than Rubio, saying "If the nominee calls you and says you can do great things for our party and our country, of course you’d listen." However, he then qualified that with "I think I’m going to be governor of Virginia for the next two years."
Chris Christie: Christie is a politician that the Intrade markets seem to rate higher than political prognosticators. Last year, he said "If the vice-presidency is offered to me, I will turn it down," and continued in that strain, "Can you see me as somebody’s vice president? I mean, who would be that poor guy? You know, I just don’t think that my personality is necessarily suited to being No. 2."
The New Jersey governor has since backed off that hardline stance. In an appearance on Face the Nation in February, Christie said, "if Governor Romney were to come and talk to me, I’d listen because I love my party enough and I love my country enough to listen."
Rob Portman: Portman is a quiet but hard-working and wonky Senator from a swing state - Ohio - where he campaigned hard for Romney. While he doesn't have the glitz that some of the other 2010 GOP freshmen do, he has a lot of other positives going for him.
Unfortunately, he closed the door on speculation pretty quickly. In March, he said simply, "No, not for me."
Allen West: The buzz has skyrocketed recently with Sarah Palin saying that West would be her preferred man for the job. In an interview last month, West said "I am always willing and ready to serve my country in whatever capacity that the American people would desire."
Other popular candidates have included Gov. Nikki Haley, former Gov. Tim Pawlenty and New Mexico Gov. Susana Martinez. They've all made pretty strong negative statements about the possibility.
"If offered any position by Gov. Romney, I would say no," South Carolina Gov. Nikki Haley told The Associated Press a day earlier.
"I've taken myself off the list," former Minnesota Gov. Tim Pawlenty said recently.
"It's humbling, but I'm not interested," New Mexico Gov. Susana Martinez said.
The VP job can sometimes be a position with not a ton of hype around it, so the sheer lack of excitement from the GOP contenders isn't too surprising. Most politicians prefer to keep their heads down and continue work in their current capacities. But when the phone rings from the GOP nominee, a politician will surely have to put a lot more serious thought into the idea than a quick one line dismissal.