Well, at first it would be very strange. Minato would be confused by this female version of his son. But as they interact more, if fem Naruto shows her familiar Naruto - like spirit, Minato would start to see her as family. He might become very protective of her in the past world, which is different from their normal father - son relationship.
During time travel, they would rely on each other more. Kakashi would have to depend on fem Naruto's unique abilities and insights that she has gained in their original time. For example, if fem Naruto has some new jutsu or a different way of thinking about problems. At the same time, fem Naruto would look up to Kakashi's experience even more as they face unfamiliar situations in the past. This could deepen their bond and make it more of a mutual admiration and friendship rather than just a student - teacher relationship.
Their relationship would likely become more equal. In the original story, Minato was more of a distant, heroic figure to Naruto. But through time travel, they can interact as peers, sharing knowledge and experiences. This would deepen their bond.
In a Naruto time - travel Minato fanfiction, Minato could be sent to the future. There, he would see how the village has changed, and might be shocked to find out about the Akatsuki and their actions. He could then work with Naruto and the others to fight against the Akatsuki using his unique skills and knowledge of the past. For example, he could teach Naruto some new jutsu that was lost over time.
If Minato time - travels, it could change a lot. For instance, if he prevents the Nine - Tails attack, Naruto would grow up in a very different environment. He might not be the underdog he was in the original story. Also, if Minato changes things during the Uchiha clan's situation, Sasuke's story would be completely different. Konoha's political situation would also be altered. There would be less animosity between different clans perhaps.
It would be very close. Naruto has always longed to know his father, so when he travels back in time to meet Minato, they would quickly bond. They'd probably start training together right away.
In these fanfictions, time travel often gives Fem Naruto the chance to fix past mistakes in their relationship. For instance, she might prevent Sasuke from leaving the village earlier. This changes the dynamic between them, as Sasuke starts to see Fem Naruto in a new light. He becomes more curious about her and her motives, leading to a deeper emotional connection as they face new challenges together in this altered timeline.
One idea could be that fem Naruto travels back in time and has to hide her identity from Minato at first. She could then gradually reveal herself and use her future knowledge to help Minato prevent some of the tragedies that occurred in the original timeline. For example, she might know about certain Orochimaru plots and stop them before they start.
The story could start with female Naruto experiencing some strange phenomenon, like a bright light or a powerful jutsu backfiring, which then transports her back in time. As she lands in the past, she immediately recognizes the era and realizes that she has the chance to save Minato.
Their relationship would likely become stronger. If they travel back in time, they already have the memories of their relationship in the future. So, Hinata might be more bold in expressing her feelings, and Naruto would be more aware of her importance to him from the start.
In a time travel situation, they would be forced to rely on each other. Naruto's positive attitude and Fem Madara's power would complement each other. They might start as reluctant allies but gradually develop a deep bond. Maybe they'd have moments where they save each other's lives in the past, which would further strengthen their relationship. They could end up being a powerful duo that changes the course of ninja history together.