Yes, there are likely some fanfictions out there with this concept. The 'Naruto' and 'Fairy Tail' (where Natsu is from) fandoms are both quite large, and fans often like to create crossovers that connect different universes in interesting ways.
The two main characters are Naruto and Natsu. Naruto has his friends like Sasuke and Sakura in his world, but in this fanfiction, his connection with Natsu is what matters most. Natsu also has his Fairy Tail guild mates, but here, his newfound brother Naruto is the key relationship. Their interactions, their differences in powers and personalities, and how they support each other would be explored through their characters.
In this fanfiction, it could start with Naruto feeling a strange connection to Natsu. As they meet, they discover their shared blood relation. Then, they embark on a journey to find out more about their family's past. Along the way, they encounter various challenges like facing off against enemies who are after their unique powers. They rely on each other's strengths, Naruto's strategic thinking and Natsu's brute force and magic, to survive and grow closer as brothers.
The plot could involve them discovering their brotherhood. Maybe they start off in different worlds and then some magical or mysterious event brings them together. They then realize they are brothers and start to explore their combined powers to fight against a common enemy.
Maybe they embark on a journey to save their respective worlds from a common threat. A powerful demon or an evil wizard is trying to destroy both the Naruto universe and the Fairy Tail world. Naruto and Natsu would travel across different dimensions, gathering allies from both worlds. Naruto would use his Rasengan and Natsu his Fire Dragon's Roar to fight off hordes of minions while they search for a way to stop the main villain.
It could work if there was a shared backstory created for them. Perhaps they both have a common ancestor or some ancient power that links them. Naruto's strong sense of loyalty and protection could be extended to Natsu as his younger brother. He could teach Natsu some of his ninja values like never giving up and the importance of teamwork. Natsu, in turn, could show Naruto the more carefree and adventurous side of the Fairy Tail world.
Well, it's quite possible. Fanfiction writers are known for their wild and creative ideas. Since both Naruto and Natsu are from popular anime, some writer might have thought of creating a story where Naruto is somehow related as Natsu's grandfather. But it also depends on how much of the respective anime universes the writer wants to blend and how they would explain such a connection in the story.
Yes, there are likely fanfictions out there with such a concept. Fanfiction platforms are full of creative and often wild ideas. You can check major fanfiction websites like FanFiction.net or Archive of Our Own (AO3). Just search for the relevant tags or keywords to find them.
Friendship is also a big theme. Naruto values his friends deeply, and Natsu is the same in his own way. Fanfictions usually explore how their friendship would develop if they met. They would likely bond over their shared love for a good fight and protecting their loved ones. For example, they might team up against stronger enemies that neither could face alone.
In addition to Naruto and Natsu, there could also be other important characters. For example, Sasuke from Naruto's world might make an appearance if there's a connection between the two worlds. From Fairy Tail, characters like Lucy could also be involved as she often teams up with Natsu. They could all interact in various ways, depending on the plot of the fanfiction.