While here at Reason earlier today he made a call for all voters to line up behind his own party, the Republican Party, U.S. News is reporting on some skylarking thoughts from some Libertarian Party higher-ups that Paul could become a fusion candidate, nominated by both parties.
Likely over Paul's own dead body (doubtless for most voters in a general election, that will be a label he'll want to distance himself from, not embrace), and despite the two named sources, National Committee Executive Director Wes Benedict and Party Chair Nicholas Sarwark*, saying it just maybe might could happen, my own read of the L.P. rank and file from years of following it is that it would be shocking indeed if a bare majority of the types of activists who troubles themselves to end up as delegates to national L.P. conventions could allow their sense of self to survive linking their Party with the GOP, even for a candidate with as many decent qualities as Paul.
That said, the prize of the L.P's presidential candidacy, which gets you enormous political value in terms of being on the ballot in generally a vast majority of states, is at the whim of usually a very small number of people (winning the support of fewer than 300 people can on occasion earn it for you, depending on the turnout at their convention) and it's hard to generalize about what they might do. Still, I'd advise you to bet against this eventuality, strongly.
Some of the electoral realities a double-booked Paul would face:
[Richard] Winger, the ballot access expert, says if Paul wins both party nominations, his name would appear twice on ballots in Connecticut, New York and South Carolina. In states including California, New Hampshire, Ohio and Pennsylvania both party names would appear under his name. Florida and Texas ballots would not show the Libertarian party name.
The possibility of Paul winning the Libertarian nomination, of course, doesn't merely depend on the enthusiasm of his supporters or if his positions are close enough to party dogma. It also depends on who else is interested in the nomination.
The party's 2012 nominee, former New Mexico Gov. Gary Johnson, is busy running a marijuana company he hopes to build into an industry leader, but may enter the race.
Johnson, a former Republican, says he'd like to seek the Libertarian nomination again—though he's not firmly committed to doing so—and says Libertarians should not nominate Paul.
I wrote back in 2002 about the possibilities of "fusion" candidates for strengthening third parties in America's political system.
*Correction: Sarwark was misidentified in the first version of this post.