Sasuke is genuinely curious when he finds Kakashi hanging out around the village with an unknown woman. Much like Team 7's game of trying to uncover Kakashi's mask, Sasuke's interest in the gossipy drama of Kakashi having a love interest draws Sasuke's natural curiosity in the personal life of his teammate. While Sasuke tries to hide his curiosity under a cool veneer of indifferent disinterest in his teacher's personal life, it's plain to see that Sasuke is just as curious as Naruto and Sakura when it comes to deciphering Hanare's role and importance in Kakashi's life.

From the genins' perspective, there is a lot of mystery enshrouding Kakashi. One might assume that the three genins' burning curiosity on details such as their sensei's love life or their sensei's real face speaks to a desire to understand their teacher more and know him better, as Kakashi is a bit of an unknown factor. Yet they don't feel comfortable enough to ask him forthright, perhaps figuring that Kakashi will be vague and evasive and that asking Kakashi anything directly is a waste of time that won't get them any answers. That, or maybe it's just the barrier of age and authority making asking Kakashi anything an intimidating prospect.

Sasuke's natural curiosity is piqued by his sensei walking around young with a strange girl, enough that it distracts and diverts Sasuke from what he was originally doing, an unusual instance of drama and gossip disrupting his usually disciplined focus.

Naruto teasingly calls Sasuke out when he wonders if Sasuke is interested too, though Sasuke returns that it's "nothing like that". But Sasuke's actions contradict his words, and it is exactly like that: Sasuke is just as interested and curious about Kakashi's mysterious "romance" with the strange woman as Naruto and Sakura are. Sasuke attempts to maintain his cool guy façade by feigning disinterest and scoffing at the romantic drama despite the fact that he's just as interested in this development as Naruto and Sakura are and he spies on the couple too.

Of the trio, Sasuke is the only one to question Kakashi's involvement with Hanare, as he comments that, "what's he doing with someone like her?" Since Sasuke is aware that Hanare is a criminal, the fact that a respectable jonin would be with a criminal seems strange to him, though it's something that Naruto and Sakura don't care about. I don't think that Sasuke's primary concern was moral judgement or concern that Kakashi is waltzing about with a criminal, just innocuous wonderment if Kakashi is actually in love with someone, especially if that someone is a criminal.

Sasuke's interest in Kakashi's possible love life denotes a more personal interest in who Kakashi is as a person. Team 7 knows virtually nothing about Kakashi's personal life, yet they evidently have a curiosity about it, and want to know their sensei better, but find Kakashi too evasive and aloof to get close to. Thus, perhaps the notion of a woman charming enough to be able to get close to the detached Kakashi seems magically and alluringly enchanting to the trio.

That Kakashi might even have a love interest is unexpected to all three of Kakashi's pupils, who are eagerly encouraging and supportive of a match between Kakashi and Hanare. Which makes one imagine that the Team 7 genin would be happily excited if their teacher did have a love interest. If Hanare stuck around, it's not hard to imagine that the three genin might have grown quite fond of her, excited at the prospect of the aloof Kakashi actually having a girlfriend.

Despite pretending that he isn't interested in that kind of thing, Sasuke has no qualms tailing and spying on Kakashi. After Sakura dreamily remarks that, "they both seem so perfect for each other", Sasuke scoffs, "Ridiculous."

Yet despite his derisive scorn at the pairing, Sasuke creeps closer with his teammates to eavesdrop on the private conversation between Hanare and Kakashi, listening intently to Hanare as she talks about the impact Kakashi had on her life. Sasuke even blushes as he eavesdrops (the only time Sasuke has blushed in the series apart from a couple moments when he was around Sakura or when he was a really small kid).

After Team 7 accidently gets caught eavesdropping, Sasuke looks frazzled at being discovered spying on the duo, then bashfully embarrassed after Hanare thanks the genin trio for congratulating Kakashi and Hanare with flowers.

That said, at the end of the episode after Hanare escapes, Kakashi covers Hanare's escape up by telling his three students that Hanare fell to her death. Sasuke doesn't look especially horrified or distraught by her death, so while was intrigued and mystified by the prospect of Kakashi having a girlfriend, he doesn't appear to be particularly attached to that concept either.