There are many possibilities. It could be a mysterious and powerful ninja that no one expected. Since it's fanfiction, the author could come up with an entirely new character who has unique abilities and a complex backstory, and that person could be Naruto's father. Another option could be an anti - hero type of character, someone who has a shady past but still has a connection with Rouge that led to Naruto's birth.
Well, it could be anyone really. It might be some random ninja that Rouge had a relationship with. Or it could be a character created specifically for this fanfiction. Maybe it could even be a character from another village who had a secret liaison with Rouge.
Well, if Naruto is the son of Rouge in fanfiction, it could change a whole lot. For one, his relationship with the village might be different. Since Rouge was from a different background, Naruto might not be as shunned as he was in the original. Also, his abilities could be different. Maybe he'd inherit some of Rouge's unique skills instead of just the Nine - Tails' power. And his relationships with other characters like Sasuke and Sakura would also be different as his origin story is completely changed.
It could also be a result of a powerful jutsu gone wrong. For instance, a forbidden jutsu that alters the fabric of reality and somehow makes Naruto become his own father within the fanfiction world. In this case, he might have to deal with the new family dynamics and the responsibilities that come with being a father in a very strange and unexpected situation.
His father could be Ash from Pokemon. Ash has had so many adventures and it would be interesting if he somehow ended up in the Naruto world and had a child with a Naruto - world character. His mother could be a random ninja who had a connection with Ash.
In fanfiction, the possibilities for Naruto's father could be many. It could be an original character created by the author. Maybe a powerful shinobi who has a connection with the female Shinigami. For example, he could be a shinobi who made a deal with the Shinigami in the past and through some convoluted plot, became Naruto's father.
It could be Hinata. Since they got married in the original series, it would be a natural choice. Their son would inherit Hinata's gentle nature along with Naruto's determination.
In some fanfictions, Minato could be Naruto's father in the context of the Kyuubi. Since Minato was Naruto's actual father in the original series and the Kyuubi was sealed within Naruto by Minato. So many fanfictions might expand on their relationship within the framework of the Kyuubi's presence.
The plot could involve Kakashi adopting Naruto. Kakashi would then be responsible for Naruto's upbringing, teaching him not only ninja skills but also life lessons. For example, he could teach Naruto how to cook simple meals, which is a basic life skill. This would show the domestic side of their father - son relationship.
Another possibility is that Naruto's 'god' parent is an actual deity from Japanese mythology. Since the Naruto world is inspired by Japanese culture, it would make sense for a kami (god) from their mythology to be Naruto's parent. For example, Amaterasu, the sun goddess. Her power could be passed down to Naruto, giving him abilities related to fire and light.
It might be a child from a small, out - of - the - way village that was attacked by bandits. Naruto stumbles upon him during a diplomatic mission and decides to adopt him. The boy could have a natural affinity for wind - style jutsu like Naruto, which could be the start of a strong bond between them as Naruto trains him.
Another possibility is that it's a result of some sort of experimental jutsu gone wrong. So there may not really be a traditional 'father' in the sense, but rather it's a consequence of Naruto's own actions in the world of ninja magic.