(CNN) - It's an idea that's increasingly getting kicked around, especially by one campaign in particular.
But following Clinton's wins in Texas, Ohio, and Rhode Island last week - a trifecta of victories that essentially salvaged her presidential hopes - the New York senator and her surrogates have repeatedly raised the issue themselves; raised it so often, in fact, that Obama made sure to shoot down the idea at a campaign event in Mississippi Monday.
"I want everybody to be absolutely clear," he said. "I'm not running for vice president. I'm running for president of the United States of America."
"This kind of gamesmanship, talking about me as VP but not ready for commander-in-chief, that's exactly the kind of doublespeak, double-talk that Washington is very good at," Obama added.
Obama isn't the only one who thinks Clinton has something up her sleeve when she raises the issue. CNN Senior Political Correspondent Candy Crowley says the Illinois senator's campaign clearly recognizes its rival benefits from the increased speculation of a joint ticket.
"They understand that this is a way for the Clinton campaign to play some politics here," Crowley said. "If you suggest to voters, who might have some question about Barack Obama's experience, but they like his message of hope and change, that they could have both, and that maybe he could get eight years of seasoning, that's the suggestion here."(Watch Candy Crowley's analysis of Dem dream ticket talk)
Obama's comments Monday followed a week in which the prospect of a joint ticket was pushed by Clinton or a member of her campaign on at least four different occasions. Clinton herself raised the issue twice - once with a CBS interviewer on the morning after her March 4 primary wins, and on Friday during a campaign stop in Mississippi.
"I've had people say, 'Well I wish I could vote for both of you,'" she said. "Well, that might be possible some day. But first I need your vote on Tuesday."
Former President Bill Clinton also weighed in on the possibility over the weekend, calling the possible ticket 'unstoppable.'
"I know that she has always been open to it, because she believes that if you can unite the energy and the new people that he's brought in and the people in these vast swaths of small town and rural America that she's carried overwhelmingly, if you had those two things together she thinks it'd be hard to beat," he said in Mississippi Saturday.
And Pennsylvania Gov. Ed Rendell, a fervent supporter of Clinton's, said on two different occasions he thinks whoever wins the Democratic race should offer the vice presidency to the other.
Gloria Borger, a senior political analyst for CNN, called the comments a Clinton "talking point."
"They're trying to say to people, you know 'wink, wink, if you make Hillary Clinton the nominee, she'll bring Barack Obama along.' It's a talking point, which is that with Hillary you get two for the price of one—if she is at the top of the ticket."
But there's no denying the Clintons recognize the strength Obama would bring to a presidential ticket. Each has attracted a distinct segment of the Democratic electorate: He consistently does well with upscale voters, independents, and African-Americans while she has performed well among women, older voters, and members of the working class.
"Thee Clintons clearly believe that Obama on the ticket would both unify the party and be a draw, particularly to those African-American voters who have been flocking to him. There is clearly some sense it would cement a Democratic victory."
So would Clinton herself accept the No. 2 slot should Obama finish the race on top?
"I don't make any predictions," Borger said.
– CNN Ticker Producer Alexander Mooney
UPDATE: Clinton said Monday all the V.P. speculation has taken on a life of its own:
"This thing has really been given a life of its own. A lot of Democrats like us both and have been very hopeful that they wouldn't have to make a choice, but obviously democrats have to make a choice, and I'm looking for to getting the nomination and it's preliminary to talk about whoever might be on whose ticket," she said while campaigning in Pennsylvania. "But I believe I am ready to serve on day one."