An often discussed potential reason for Tywin's disgust with Tyrion's whoring is that it reminds him of himself. With that in mind I wanted to look around and see if Tywin has any potential bastards out there.

[Were there any bastards of Tywin Lannister?]

GRRM: He'd have you flogged for the mere suggestion. -SSM, Various Questions: 15 April 2008

A Look into any Potential Bastards of Lord Tywin Lannister

King's Landing

We know that Tyrion used a tunnel in the Tower of the Hand to get to Chataya's brothel:

"How is it a brothel happens to have a secret entrance?"

"The tunnel was dug for another King's Hand, whose honor would not allow him to enter such a house openly. Chataya has closely guarded the knowledge of its existence." -ACOK, Tyrion III

and this is what GRRM has had to say about the tunnel in the past:

[Was Tywin the Hand who made the secret tunnel into Chataya's, to hide his visits?]

GRRM: Interesting theory. -SSM: Asshai Forum Chat

and:

For which Hand was dug the passage to Chataya's?

GRRM: No comment. Feel free to theorize...- SSM Monthly Correspondance

Using that to get to Chataya's and looking at the possibilities the best option (imo) is Marei

• Marei

Marei was a cool, pale, delicate girl Tyrion had noticed once or twice. Green eyes and porcelain skin, long straight silvery hair, very lovely, but too solemn by half. "I'd hate to have the poor child lose her pearls on account of me." -ACOK, Tyrion VII

Since Marei is older than Shae (who is around 17 in 299AC), it likely puts her at an age where she was conceived when Tywin was Hand of the King (262 to 281 AC)

That said we do know that Tywin loved Joana (but that doesn't mean he refrained from whores).

The Westerlands

• Donnel Hill

The personality seems a bit more like Gerion and he has fat lips like Joffrey and yellow locks, but still had to include him:

Of the dozen odd brothers who sat by the fire, four were his. He gave each one a hard squinty look as he ate, to see if any showed signs of breaking. Dirk seemed calm enough, sitting silent and sharpening his blade, the way he did every night. And Sweet Donnel Hill was all easy japes. He had white teeth and fat red lips and yellow locks that he wore in an artful tumble about his shoulders, and he claimed to be the bastard of some Lannister. Maybe he was at that. Chett had no use for pretty boys, nor for bastards neither, but Sweet Donnel seemed like to hold his own. -ASOS, Prologue

• Caspor Hill

He is only mentioned in the ADWD, Appendix and in passing:

Ser Franklyn did the introductions. Some of the sellsword captains bore bastard names, as Flowers did: Rivers, Hill, Stone. Others claimed names that had once loomed large in the histories of the Seven Kingdoms; Griff counted two Strongs, three Peakes, a Mudd, a Mandrake, a Lothston, a pair of Coles. Not all were genuine, he knew. In the free companies, a man could call himself whatever he chose. By any name, the sellswords displayed a rude splendor. Like many in their trade, they kept their worldly wealth upon their persons: jeweled swords, inlaid armor, heavy torcs, and fine silks were much in evidence, and every man there wore a lord's ransom in golden arm rings. Each ring signified one year's service with the Golden Company. Marq Mandrake, whose pox-scarred face had a hole in one cheek where a slave's mark had been burned away, wore a chain of golden skulls as well. -ADWD, The Lost Lord

War of the Ninepenny Kings

Tywin fought in the War of the Ninepenny Kings (at age 18). Its possible that a situation like this could have happened in the Stepstones (not arguing for Chain explicitly, just pointing out the possibility for members of the Golden Company, etc.):

Chain himself had been born into the company, fathered on a camp follower by his sellsword father. Though he had been raised to speak the Common Tongue and think of himself as Westerosi, he had never set foot in any part of the Seven Kingdoms till now. -TWOW, Arianne II

These were just the most likely options I could find, but the possibilities are pretty endless.

TLDR: Tywin Lannister may have fathered a few bastards around the realm