This was a short novel (maybe 200 to 250 pages) that I read in the 1990s, but which based on style I would think was published in the 70s, and almost certainly in the late 60s to mid-80s period.
The protagonist is a new doctor in a hospital who takes a special interest in a long-term patient that has been there for decades. The patient, a man, is in a catatonic state. I think that the patient was able to make some voluntary movements, so he could cooperate minimally with his care and feeding. The doctor discovers that the patient has never had a diagnosis that ascribes any cause to the man's condition and makes an effort to get to the bottom of the case. If I'm recalling correctly, the man appears to be younger than the records show that he is.
The doctor becomes somewhat obsessed with the matter over the course of weeks or months, learning very odd details such as that the patient always eats and drinks exactly the same amount every day, and at exactly the same times. I think he further discovers that all bodily functions (heartbeat, respiration, and even excretion) are similarly precise and unvarying. He may also start to notice that other people behave somewhat oddly around this patient, not really thinking about the man directly very often but sometimes compelled to do things like open the shades in his room or the like. He's not able to interest other doctors in the case, even the ones that should be.
My memory gets a little hazy here, but I think the climax is that the doctor begins to have telepathic contact with the patient, who is some kind of superbeing mentally roving the universe while his body is on autopilot back on Earth. I know that the doctor attempts to prove this to his colleagues with an extensive presentation detailing all of his odd findings, but they don't believe him. It is either shortly before or shortly after this that the superbeing makes telepathic contact.
The story starts to take on an increasingly odd tone over time, such that it's reasonable to interpret the culmination of the plot as the doctor suffering a nervous breakdown, and that he is wholly deluded about this patient. I don't recall how the story ends.
For some reason I was thinking that this was a work by Frederik Pohl, but a thorough check of his bibliography is turning up nothing that looks right.
EDIT: Despite some plot similarities, this is definitely not "The Strange Case of John Kingman" by Murray Leinster, in which the patient in question is an extraterrestrial. The patient in this story was definitely human, had a birth record, and (if I'm recalling correctly) had been placed into institutional care relatively early in his life according to the protagonist's research.
I am not familiar with a short story version of the story, but it is quite possible that the book I read was an expansion of a short story or novella, given its short length.