One possible consequence is that he might become more confident earlier. With the Rasengan in his arsenal sooner, he could face opponents more bravely in the early stages of his ninja journey. This newfound power could also make him a target for stronger enemies who see him as a potential threat.
One possible plot could be that Naruto discovers an ancient scroll that has a simplified version of the Rasengan technique. He spends days secretly practicing it in the forest, facing initial failures but gradually getting the hang of it. With his new - found skill, he surprises his friends and even manages to use it in a sparring match against Sasuke, changing the dynamics of their rivalry early on.
It could be Kakashi. Even though in the original story he doesn't teach Naruto the Rasengan, in a fanfic he might have some hidden knowledge about the technique. Maybe he was secretly studying it himself and saw Naruto's eagerness to learn new jutsu, so he decided to share his knowledge and help Naruto learn the Rasengan earlier.
His training would likely be more focused on perfecting the Rasengan. Instead of spending time on basic jutsu training, he might be pushed to improve the Rasengan's power and speed. For example, he could be taught different ways to combine it with his other skills.
In many fanfictions, Naruto might master the Rasengan through intense and unique training methods. Maybe he gets some special guidance from an unexpected source, like a hidden shinobi in the village who has a different perspective on chakra control. This person could show Naruto some new ways to focus his chakra, which finally enables him to form the Rasengan successfully.
In some fanfictions, Kakashi could also play a role. He is very knowledgeable and has seen the Rasengan in action. He might offer some tips on chakra control, which is crucial for Naruto to master the Rasengan.
Maybe Sasuke teaches him the Chidori in a moment of camaraderie. And for the Rasengan, it could be an older and wiser Naruto from the future who travels back in time to teach his younger self.
With Shikamaru, Naruto's early Rasengan might make Shikamaru more interested in teaming up with him. Shikamaru is smart and he would recognize the potential of having Naruto with his powerful jutsu on their side for missions. So they could form a stronger bond based on mutual respect for each other's abilities.
One creative version could be where Naruto learns Rasengan in a dream realm. In this fanfiction, the dream realm has unique chakra rules. He meets a mysterious sage who teaches him a faster way to form the Rasengan. Another could be a time - travel fanfic where Naruto goes back in time and learns Rasengan from his father. It gives a whole new perspective on the jutsu and his family connection. And there could also be a version where Naruto learns Rasengan from an enemy turned ally, which shows his ability to find common ground even with foes.
The key element is Naruto's determination. He has to be really determined to learn both the Rasengan, a powerful chakra - based attack, and the Hiraishin, a complex space - time ninjutsu. Without that strong will, he wouldn't be able to endure the tough training.