One possible scenario in this fanfiction could be that Naruto grows up with a more balanced upbringing. Tsunade might teach him medical ninjutsu and how to control his emotions better. Jiraiya, on the other hand, would surely train him in powerful combat techniques and in the ways of the sage. Naruto would probably be more mature and well - rounded at an earlier age, with a combination of Tsunade's tough love and Jiraiya's adventurous spirit guiding him.
Naruto may also develop a darker side to his personality. The abandonment could make him feel betrayed and angry. He might struggle with controlling his emotions, especially his anger. However, this could also lead to him learning how to channel that anger into power in a more controlled manner, like he did with the Nine - Tails' chakra, but in a more refined way.
One possible plotline could be Naruto feeling extremely lost and having to find his own way to become stronger without their guidance. He might start to question his own worth and abilities, but then he could meet new allies who help him realize his true potential on his own. For example, he could encounter some other outcast ninja who have their own unique skills and together they form a new kind of ninja team that challenges the traditional ways of the ninja world.
In this fanfiction, their relationship would be transformed. They would be like a married couple in a way, but without the formalities. Jiraiya would probably tone down his womanizing ways as he has a family to think about now. Tsunade would be more open and less prickly around Jiraiya. They would bicker less and cooperate more. Their relationship would be centered around Naruto, and as he grows, they would grow together as a family unit, with their bond getting stronger over time.
In the official Naruto story, Naruto is the son of Minato and Kushina. However, in the realm of fanfiction, it's possible for such a concept to exist. Fanfiction writers often come up with creative and alternative storylines. But this goes against the original Naruto lore. It would be a very different take on the Naruto universe, with implications for Naruto's character development, his relationships with other characters like Sasuke and Sakura, and the overall plot. For example, his training might be influenced by Tsunade and Jiraiya in a more family - oriented way rather than just as mentors.
He would probably also feel angry. After all, he has sacrificed so much for the village and for them. He might start to question his own beliefs and values. But Naruto being Naruto, he would not give up easily. He would try to prove his innocence and show them that they are wrong. He could go on a journey to find evidence or seek help from other allies who still believe in him, like Shikamaru or Kakashi.
He could also try to talk to them directly, despite the betrayal. Naruto has always believed in the power of communication and the bonds between people. He'd want to understand why they thought he was worthy of betrayal and try to convince them that they were wrong.
One great fanfic could be about their adventures before Naruto was born. It might explore their relationship dynamics, how they trained together and supported each other. Maybe in this fanfic, Jiraiya tries to convince Tsunade to take on more dangerous missions with him, showing his trust in her abilities.
No, in the official Naruto series Naruto is not related to Tsunade and Jiraiya in that way. But in fanfiction, writers can come up with all kinds of wild ideas. They might write a story where Tsunade and Jiraiya had a relationship that resulted in Naruto. This could lead to interesting plot twists, like Naruto having different powers or a different personality. Maybe he would inherit Tsunade's super strength and Jiraiya's toad - related jutsu. But remember, this is all just fanfiction and not part of the original Naruto plot.
One possible plot could be that Tsunade and Jiraiya are deceived by false information about Naruto. Maybe they are made to believe that Naruto has turned to a dark side. For example, an enemy might have created false evidence of Naruto collaborating with a powerful evil force. Tsunade, being the Hokage, and Jiraiya, Naruto's mentor, would then be forced to take actions against Naruto that seem like betrayal from Naruto's perspective.