One possible reason could be that Kakashi might think Naruto is not ready yet. Maybe Naruto hasn't mastered some basic skills that are prerequisites for the more advanced training Kakashi has in mind. Another reason could be that Kakashi is worried about Naruto's recklessness. Naruto has a tendency to rush into things without thinking, and Kakashi might fear that this could lead to Naruto getting seriously injured during the training.
Maybe Kakashi is preoccupied with other important missions. He is a highly - skilled and often - needed ninja. There could be some top - secret missions that require his full attention, leaving him no time to train Naruto properly.
In the fanfic, Kakashi might neglect Naruto's training due to a variety of reasons. He could be preoccupied with his own research or a secret mission that demands his full attention. There could also be a misunderstanding between him and Naruto. Maybe Naruto has done something that Kakashi misinterprets, causing him to distance himself from Naruto's training. Additionally, the fanfic could introduce new characters or plot elements that draw Kakashi's focus away from Naruto. For example, a new threat to the village that Kakashi feels he must handle alone, thus neglecting his duties as Naruto's trainer.
Another reason could be that in this fanfiction, the plot is set in a way that there is a new training system in the village. And this system doesn't involve Kakashi in Naruto's training. There could be new mentors or a completely different approach to training young ninjas like Naruto, which excludes Kakashi from the equation.
One reason could be that he saw Naruto's potential later on and realized he could have been much stronger if he had trained him earlier. For example, Naruto has a great deal of chakra and tenacity, and with proper training from Kakashi earlier, he might have reached new heights faster.
In the 'naruto denied training fanfiction', perhaps Naruto denies the training because he doesn't trust the person offering it. Naruto has had his fair share of enemies and untrustworthy characters. If the trainer is someone he doesn't know well or has a bad feeling about, he would surely turn down the training. Also, he might be confident in his own self - taught skills and the training he has received from his masters like Jiraiya.
Naruto might initially be very disappointed. He has always been eager to train and become stronger. So being denied by Kakashi could make him feel a bit down. But he could also be defiant. He might start training on his own, determined to show Kakashi that he doesn't need his approval to get stronger.
In the fanfiction, Kakashi might train Naruto in teamwork. Even though Naruto is often a solo fighter, Kakashi could show him the importance of working with others. He could arrange mock battles where Naruto has to team up with other characters, like Sakura or Shikamaru, and learn how to coordinate their attacks and defenses.
In many Naruto fanfictions about Kakashi training Naruto, Kakashi might push Naruto to new limits. He could focus on Naruto's Rasengan training, making it more powerful or teaching him new variations. For example, Kakashi might introduce elements to the Rasengan, like wind or lightning, to enhance its destructive power. This kind of training would not only strengthen Naruto physically but also help him understand chakra better.
One possible result is that Naruto becomes much stronger in a shorter period compared to the original story. His combat skills improve significantly, allowing him to take on more difficult missions. For example, he might be able to defeat enemies that were previously out of his league with ease.
Naruto might also try to find alternative ways to gain strength. He could explore the forest around the village to find new training areas or discover new chakra - related phenomena. He has a curious nature, and this could lead him to stumble upon something that gives him an edge in his training, like a hidden hot spring that has special chakra - enhancing properties.