It has been occasionally proposed that there is reason to doubt Tyrion Lannister’s paternity. Items such as his extremely pale blond hair, which stands in contrast to the golden blond of his family, have been made note of, as has Tyrion’s fascination with dragons. Some also note that his statement in A Storm of Swords suggests Tywin has doubts as to Tyrion’s paternity: “Men’s laws give you the right to bear my name and display my colors, since I cannot prove that you are not mine.” Finally, the prophecy of the dragon having “three heads” has been taken to mean that Tyrion is one of the three heads, and so a future dragon rider, necessitating his having Targaryen blood.
A Dance with Dragons does provide a thread of support for the theory that Tyrion is not Tywin’s. Barristan Selmy reports the following to Daenerys:
“Prince Aerys . . . as a youth, he was taken with a certain lady of Casterly Rock, a cousin of Tywin Lannister. When she and Tywin wed, your father drank too much wine at the wedding feast and was heard to say that it was a great pity that the lord’s right to the first night had been abolished. A drunken jape, no more, but Tywin Lannister was not a man to forget such words, or the . . . the liberties your father took during the bedding.”
The common speculation is that Aerys Targaryen raped or seduced Joanna Lannister at some point, leading to Tyrion’s birth and Joanna’s death, and this at least provides proof that Aerys was interested in Joanna. However, the objections are obvious. Why did Tywin continue to serve as Aerys’s Hand if he suspected this? So far as Tywin or anyone else ever indicates, his departure from the role of Hand and his fateful break from Aerys were the result of long-building tensions thanks to the king’s madness and his stealing away of Ser Jaime to serve in the Kingsguard. For that matter, why did he persist in trying to wed his daughter to Aerys’s heir, if he had cause to hate Aerys so much?
The response to this might be that Tywin suspects Tyrion’s paternity but does not have Aerys (or, perhaps, anyone) in mind as the culprit, or perhaps that he learned about this fact only long after the fact. The latter seems a weak counter, as it hand-waves the details and simply leaves a nebulous possibility that can’t be rejected but really has no evidence for it. The former, however, seems a better objection—though again, unprovable, especially now as Tywin is dead.
On a metatextual level, it seems as if there’s much greater force to having Tyrion in fact be Tywin’s son—and thereby maintain his status as a kinslayer, among other things—than to reveal that in fact he is a secret Targaryen bastard, especially as it seems the narrative has already provided not one, but two, secret Targaryen offspring (see the FAQ entries on Jon Snow and Young Griff).