Villain Takes an Interest

So you have a pair of Arch Enemies: a hero and a villain. And then...the villain gets interested in the hero.

The hero and villain are very similar to each other, and both grow aware of that fact. But they have different reactions. While the villain genuinely wants to take the hero under his or her wing, the hero is opposed to any friendly interaction—but may fear the villain has a point. The kind of obsessive behavior and mind games this generates on both ends is a prime source of Foe Yay.

If the hero and the enemy do end up becoming mentor and student, the ensuing relationship can range from the worst kind of Training from Hell to a downright parental role. Both will still probably end up odd. And it'll end badly for at least one of them.

The Hero will usually consider The Mentor to be their Arch Enemy and be very personally devoted to taking them down; The Mentor character is typically more amused by this than anything, and may take the role of a Stealth Evil Mentor. Foe Yay may lead to a Mind Game Ship, a Mentor Ship, or both.

Related to We Can Rule Together, but rather than offering a position, offers a study period. Note that this requires a mentoring role--Worthy Opponent offering a team-up doesn't cut it.

Examples of Villain Takes an Interest include:

Anime and Manga

Gauron/Sousuke in Full Metal Panic!.
Kohei Kakihara and Chiko from The Daughter of Twenty Faces. Kakihara is fascinated by her fighting spirit. His obsession to take away everything Twenty ever had may have something to do with this as well.
Kyosuke to Kaoru in Zettai Karen Children. Incidentally, Kaoru also has a regular mentor, and the ultimate question of the plot is whether Kaoru will ultimately side with Minamoto or Kyosuke.
For the dubious amount of villainy, Evangeline after defeating Negi says she won't kill him, because she found interest in him, but it's only when he ask her to teach him (after witnessing her power on a more serious level) that they become mentor and student.

Comic Books

In the Ultimate Spider-Man line, the second Spider-Man, Miles Morales, is approached by his uncle, the super thief Prowler, for training. Little does he know, the Prowler needs Spider-Man to deal with the extremely powerful Maximus Gargan—the Scorpion.
Some tellings of Spider-Man give him this sort of connection to Norman Osborn, aka the Green Goblin. Especially when the Green Goblin's son is not living up to his father's expectations.
In the Green Lantern comics, Sinestro has taken an interest in Soranik Natu, the current Lantern from his home planet of Korugar. Where Sinestro was hated for being a Knight Templar tyrant (and the next Korugarian Lantern, Katma Tui, was hated by proxy), Korugar is coming to love Soranik. Both are viewing the peoples' growing devotion to her as practically asking her to be a dictator herself, which Sinestro views as her bringing order where he can't anymore while Soranik is greatly troubled by it. And it's no help at all that Sinestro is Soranik's biological father.

Film

Star Wars:
- Luke Skywalker and Darth Vader have strong elements of this, minus the Foe Yay because, well, squick. The fixation leads to both Last Second Chance and We Can Rule Together moments. Of course, from Vader's perspective, he is neither villainous nor beginning an interest.
- Also the Emperor toward Luke.
Treasure Planet does this, especially when it's becoming apparent Silver was Evil All Along. Subverted past the climax, however, when Silver slips into a somewhat benevolent role and loses his edge.

Video Games

Terra and Kefka in Final Fantasy Dissidia.
In the Fan Sequel The Silver Lining, Graham has a strange vision where Manannan claims his real plan had been to treat Alexander as an apprentice and surrogate child. Alexander's reaction to the revelation is a combination of squick/horror/disbelief.

Western Animation

Robin and Slade from the Teen Titans cartoon. The similarity is emphasized whenever they fight each other, and especially when they fight together, as their fighting styles are nearly identical.
- To a somewhat lesser extent, this is observable between Cyborg and Brother Blood as well.
- Slade and Terra have a bit of this, too. Terra allows him to mentor her and...things go downhill.
Danny and Vlad from Danny Phantom. Vlad is fixated on Danny's mother, Maddie, and ultimately hopes to steal her away from Jack. Add the fact that Vlad and Danny are pretty much the only half-ghosts in existence, and Vlad sees Danny as the perfect Child Substitute. Naturally Danny doesn't see it this way.
Rex and Van Kleiss from Generator Rex. Although it seems that he is more interested in Rex's nanites than the boy himself.
- Van Kleiss himself supplied that he would be happy if Rex was a vegetable as long as he was still breathing. Unlike some other examples, Rex is Genre Savvy enough to take advantage of this without entering such a relationship. In the episode "Alliance" Rex uses guile and half-truths to undermine Van Kleiss in various ways. Most notable of them is getting Circe to defect from the Pack. Circe was convinced that Van Kleiss wants to take Rex in like he did with her. So when Rex is forced into an arena with No-face for a No-Holds-Barred Beatdown, Rex makes it clear to the onlooking Circe that Van Kleiss places no value on him other than his life.
Omi and Chase Young from Xiaolin Showdown.