In season six's second episode, Tyrion ventures into the bowels of Meereen to scope out Dany's two dragons, Rhaegal and Viserion. They're locked up, starved, and pissed off, to say the least, so approaching them isn't exactly wise. (They roast a Meereen noble who gets too bold to prove this point.) But Tyrion is able to have a chat with them and free them from their chains, which immediately had fans wondering if he'd ever ride them in the future.
Thus far there hasn't been any indication that he'll be the third rider in the prophecy. The only one who's gotten close to the dragons other than Daenerys herself is Jon. However, there's been speculation that Tyrion is actually a Targaryen rather than a Lannister that further supports this.
Although Tywin raised Tyrion as his own (albeit begrudgingly), a number of fans believe that King Aerys II — a Targaryen — raped Tywin's wife, Joanna, on their wedding night, which supposedly resulted in baby Tyrion, who Joanna died giving birth to. There's also the fact that Tyrion is described as having a number of Targaryen traits in the book (hair so blond that it verges on white).
On top of the dragons' clear acceptance of Tyrion, it's mentioned that Tyrion is fascinated with dragons in general. In the scene with Rhaegal and Viserion, he tells a story about how he begged his uncle for a dragon on his Nameday as a child, and the books hint that Tyrion has had dragon dreams (which is something Targaryens frequently experience).