Maybe it's the high expectations that Naruto has for Boruto. Naruto is a hero, and he might unknowingly put a lot of pressure on Boruto to follow in his footsteps. Boruto could be scared of not being able to meet those expectations.
Boruto might be in awe. He has always known his father as the Hokage, a powerful and respected figure. Seeing Naruto's past full of struggles, loneliness, and determination would make him realize how far his father has come. It could also make Boruto more empathetic towards Naruto and perhaps inspire him to be more like his father in some ways.
In a 'naruto dies boruto fanfic', Boruto's character would go through a major transformation. He would be forced to grow up overnight. Initially, he might be consumed by anger and a desire for revenge if Naruto was killed unjustly. But as time passes, he would learn to channel that energy into positive change. He would study his father's leadership style and try to implement it in his own way. He would become more diplomatic, forging alliances with other villages to strengthen Konoha's position. His sense of justice would be refined, and he would be more committed to protecting the world his father loved so much.
It could also be that Naruto sees in Boruto's eye a sign of an emerging danger. The eye might hold a clue to a new enemy or a brewing conflict. Naruto, being the experienced ninja he is, would then start to take precautions and train Boruto more intensively to face whatever this sign in his eye indicates.
Seeing Naruto's power in fanfiction can greatly influence Boruto's character. It could make him more determined to reach his father's level. He might start training harder and be more serious about his ninja duties.
They could start by having an honest conversation. Naruto could sit down with Boruto and just listen to his fears without judgment. This would be a first step in building a better relationship.
One great 'Boruto and Naruto fanfic' is 'A New Generation's Bond'. It beautifully shows the relationship between Boruto and Naruto as they face new threats together. The characterizations are on point and the story has a good flow.