One of the popular characters paired with Sarutobi is Jiraiya. They are both important figures in the Hidden Leaf and their relationship, whether it's as comrades or as teacher - student in some fanfic interpretations, is always interesting to explore.
In a Naruto saves Asuma fanfiction, Naruto could use his new jutsu skills. Maybe he discovers a way to reverse the damage done to Asuma. It could be related to his Sage Mode powers, which he could use to heal Asuma's wounds.
Asuma's training might make Naruto more disciplined. Naruto is known for his impulsive nature, but under Asuma's guidance, he could learn to think before acting.
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.
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.
Yes, there are quite a few. You can start by looking on popular fanfiction websites like FanFiction.net or Archive of Our Own. Some of them might focus on their unique bond, like adventures they could have had together or how Asuma's teachings could have influenced Naruto in different ways.
One interesting Naruto Asuma fanfiction could be about his unfulfilled dreams. It might show how he had plans for the future of the village and his students that were cut short by his untimely death. It could also include his thoughts on important events in the Naruto world that he didn't get to fully participate in. Fanfictions often give a chance to explore these 'what - if' scenarios.
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.