According to Naoto's winquote against Terumi, he's got a presence of someone who is associated with the Terumi clan. The Terumi clan's leader has the Hunter's Eye - a special ability in their right eye that allows them to see the life force values. After Saya Terumi killed all of her clan except for Naoto, Naoto was selected by the Hunter's Eye as the head of the clan. When Saya tried to kill Naoto in the second novel, she said that she'll take his eye.
Terumi, in his ghost form, doesn't have a right eye. And "yūki" means "ghost". But, according to Mori, Terumi's name is not his real name. But what if Mori was talking only about the "ghost" name?
I personally don't think Terumi is part of the Terumi clan, therefore making him unable to have Hunter's eye as well. I did thought about it once. Terumi is mostly likely not even his real name after all. But it's a theory, it can be true or false. I like to point out that Terumi in his "human" form has his right eye revealed most of the times while his left eye glows green (his CP Select Artwork).