One way is through joint training sessions. Sarutobi Sensei could push Naruto to his limits and in the process, they would learn more about each other's strengths and weaknesses. Also, they could be put in a situation where they have to rely on each other completely, like being trapped in a dangerous genjutsu. Sarutobi Sensei could then use his wisdom to guide Naruto out, strengthening their relationship. Another aspect could be Sarutobi Sensei introducing Naruto to important figures from his past, which would give Naruto more context about his sensei and bring them closer.
One interesting storyline could be Naruto learning some ancient and forgotten jutsus from Sarutobi Sensei. This could involve Sarutobi Sensei passing on his vast knowledge of unique techniques that were lost over time. Another could be a time - travel story where Naruto goes back in time and meets a young Sarutobi Sensei, and they team up to face a common enemy. It could also be a story where Sarutobi Sensei trains Naruto in the art of leadership, much like he did for the previous generations in Konoha.
Perhaps in this fanfiction, Sarutobi is trapped in a genjutsu created by a rogue ninja. Naruto, being highly resistant to genjutsu due to his Uzumaki blood, manages to break into the genjutsu world. There, he has to fight his own inner demons as well as the rogue ninja's illusions to free Sarutobi.
I really like 'Naruto, Sasuke and Sarutobi: A New Beginning'. In this fanfic, it starts with Sarutobi having a premonition about the future of Naruto and Sasuke. So he takes steps to guide them differently. It's an interesting take on their relationship. 'Sasuke's Redemption under Sarutobi' is also good. It shows Sasuke trying to make amends with the village with Sarutobi's guiding spirit in mind. And 'The Hidden Power of the Trio' is a fanfic that gives a new perspective on the powers they possess and how they can be combined for the greater good.
In this type of fanfiction, it's possible that the story delves into Naruto's backstory and how being Sarutobi's son would have affected his childhood. Maybe he would have grown up with more guidance and support, which could have changed his personality. Instead of being the lonely outcast that he was in the original story, he could have been more confident from the start. The fanfiction might also introduce new characters related to Sarutobi that play important roles in Naruto's life. There could be new adventures and challenges that are unique to this new family dynamic, such as Naruto having to uphold the Sarutobi family name and legacy within the Hidden Leaf Village.
Perhaps he uses some ninja techniques like a shadow clone jutsu. He could make multiple clones to help him sort through and complete the paperwork quickly.
In the fanfiction, Naruto's new sensei could be a mysterious figure who has ties to the Sage of Six Paths. This person might have been waiting for the right time to appear and train Naruto. They could possess knowledge about the origin of chakra and how to fully unlock Naruto's power. It could also be someone who was once an enemy but has now changed their ways and wants to use their skills to train Naruto for the greater good of the ninja world.
There's a chance that Naruto's new sensei could be a self - taught ninja who has developed their own unorthodox style. This person sees Naruto's ability to think outside the box and wants to refine that in him. They could introduce Naruto to a new way of looking at chakra control and combat strategies that are completely different from what he has been taught so far.
Naruto would probably be extremely hurt. He has always looked up to Sarutobi as a leader and a father figure. His first reaction might be disbelief, unable to accept that Sarutobi would do such a thing. Then he would likely feel a great sense of anger and betrayal. He might lash out and try to prove his innocence.