Yes, there are likely fanfictions out there with this concept. The Naruto fandom is vast and creative, and crossovers or unique concepts like Naruto having White Zangetsu are not unheard of. You can start by searching on popular fanfiction websites.
Perhaps in this fanfiction, Naruto is given White Zangetsu as a gift from a mysterious figure. With it, he discovers new powers and abilities. He might have to train hard to master it, and in the process, he could face challenges not only from his own world but also from forces that are attracted to the power of White Zangetsu. This could lead to new alliances and enemies for Naruto, completely changing the dynamics of the Naruto universe.
One plotline could be that Naruto finds Zangetsu during a mission in a hidden part of the ninja world. He then has to keep it a secret from his village while he tries to figure out how to use it. Another could be that an enemy uses some sort of dark magic to force Zangetsu onto Naruto, and he has to learn to control it to break free from the enemy's control.
One possible plot could be that Ichigo, after fusing with Zangetsu, gains new and enhanced powers. He might have to face a powerful new enemy that only his fused form can defeat. Maybe he struggles at first to control the combined power, causing some chaos in the Soul Society.
Perhaps in a fanfic, Naruto travels to a different realm where the spirits of Zangetsu and Hyourinmaru exist. He has to pass certain trials and challenges set by the spirits themselves. If he can prove himself worthy, they decide to bond with him and become his weapons. This would involve Naruto showing his courage, determination, and his ability to use their powers for good rather than for selfish reasons.
It could be really interesting. Naruto is known for his strong will and determination. If he had Zangetsu, he would likely try to master it quickly. Since Zangetsu has a mind of its own in the Bleach series, there could be some fun interactions between Naruto and the sword spirit. He might even use Zangetsu to cut through some of the toughest jutsu barriers or opponents that he couldn't defeat with his normal techniques.
Rukia would probably be both amazed and a bit worried. Amazed at the new power Ichigo has, but worried about how it might change him. Chad would see it as a natural progression of Ichigo's power and be supportive. Orihime might be a bit scared at first, but then she would see the good in Ichigo's new form and try to encourage him.
Yes, in fanfic, Naruto can have Zangetsu and Hyourinmaru. It would be an interesting concept. Zangetsu is from Bleach and Hyourinmaru as well. If Naruto had them, his combat abilities would be greatly enhanced. He could use Zangetsu's unique sword forms and Hyourinmaru's ice - based powers in combination with his own ninja skills. This would make for some really cool fight scenes against tough opponents like Madara or Kaguya.
The relationship can start with respect. Since Zangetsu is a powerful entity, Ichigo would initially respect her strength. As they interact more, like during training sessions, they can start to share personal stories. This sharing can lead to a deeper emotional connection.
Perhaps in the fanfiction, Female Zangetsu has been separated from Ichigo due to some dark magic. Ichigo then has to embark on a journey to find her. Along the way, he discovers more about her past and her true nature, which deepens his love for her when they are finally reunited.