Like everyone else, Rantaro can be a bit annoyed with Kokichi's behavior. While the two are not shown interacting much, Kokichi called him "My dearest Rantaro" ("My beloved Amami-chan" in the original Japanese) and cried when he saw his corpse. However, his attitude changed very quickly, making it hard to tell which one of his reactions was truthful. Kokichi later claims he doesn't want Rantaro to come back to life, because he'd rather have Shuichi's wish to have Kaede come back true, to make Shuichi happy.
Much later, it's seen that Kokichi has the life-like statue of Rantaro made by Angie hanging upside down next to his bed, as a part of the evidence pieces he has hoarded. He chose it over Kaede, Kirumi and Ryoma's ones, indicating that he has more interest in Rantaro than them.
During one of the anthology stories, it's shown that Rantaro might have a big brotherly attitude towards Kokichi. When the students are attempting to catch Kokichi, they set an "Amami trap", with him waiting in place and attempting to catch him with his arms. However, Kokichi sweetly asks him to let him go and calls him "big bro", causing Rantaro to immediately soften and let him pass "just for that one time", to the annoyance of the other students. He is seen treating Kokichi somewhat similarly in other stories as well, treating him more kindly and patiently as if he were a child, though in his thoughts he is finding Kokichi quite bothersome and a type of person he will probably never get used to.
One of the reasons that made some fans ship Rantaro and Kokichi is the latter having Angie's wax figure of Rantaro in his room.
Another reason came from an official art (the page's info-box picture) showing the two together in a rather suggestive pose.
- Rantaro Amami/Kokichi Oma tag on AO3
- Rantaro Amami & Kokichi Oma tag on AO3
- Oma/Amami tag on FanFiction.Net