Based on Carl G. Jung’s typology [Jung, 1921], people can be classified using four mental functions sensing, intuition, thinking and feeling together with the attitude (extraversion-introversion). Jung was fond of saying that personality types often change throughout ones life. So an answer to your question will be more or less inaccurate, depending upon which part of his life is under consideration. I find it understandable that several different personality types have attribute to Jung : INFP, INTJ, INFJ, INTP. Jung identified himself as both  ISTP (when he was young) and 1959 as INTP. I take the late C.G. Jung. There is almost consensus  that Jung was an iNtuitive Introvert, and thinking instead feeling but what is judging-perceiving preferences. Myers identifies Jung as an introvert. Keirsey & son identify Jung as INFJ. Von Franz identifies Jung as IN-J. 

There is a funny story about an Introvert and iNtuition told by C.G. Jung. about a “lady with a snake in the abdomen” who lived in a private brothel without noticing it (BBC Interview Face to Face available in youtube). These individuals, Jung thought, are profoundly influenced by their internal motivations even though they do not completely understand them. They find meaning through unconscious, subjective ideas about the world. Introverted intuitive people comprise a significant portion of mystics, surrealistic artists, and religious fanatics. Pretty often they are mystic dreamers or shamans or seers. After World War I, this 28-year-old woman consulted Jung. She wanted “to be cured within ten hours” – that is, within only ten analytic sessions. The woman told Jung that “she had a black serpent in her belly.” That was the reason the woman had consulted him, “for she thought that it should be awakened” . The woman was “only intuitive, entirely without a sense of reality.” Then she announced that the snake, which had been dormant, had suddenly become active. “One day she came and said that the serpent in her belly had moved; it had turned around,” Jung says. “Then the serpent moved slowly upward, coming finally out of her mouth, and she saw that the head was golden.” Jung amplifies the image of the snake in the abdomen by reference to the serpent in Kundalini Yoga. “I told you,” Jung says, “the case of that intuitive girl who suddenly came out with the statement that she had a black snake in her belly.” He situates the snake in the context of the collective unconscious. “Well now, that is a collective symbol,” he says. “That is not an individual fantasy, it is a collective fantasy.” The image of the snake in the abdomen, Jung says, “is well known in India.” Although the woman “had nothing to do with India” and although the image “is entirely unknown to us,” he says that “we have it too, for we are all similarly human.” When the woman first told Jung about the snake in her belly, he wondered whether “perhaps she was crazy,” but then he realized that “she was only highly intuitive.” She had intuited a typical, or archetypal, image. “In India,” Jung says, “the serpent is at the basis of a whole philosophical system, of Tantrism; it is Kundalini, the Kundalini serpent.”

Was Jung an INFP?

Of the four types, INFP seems the least likely type for Jung. While capable of seeing the big picture and valuing holism to a certain extent, Jung was far too analytical, systematic, and attentive to detail to qualify as an INFP. He himself saw his feeling function inferior the 1959 interview below.

Was Jung an INTP?

This is most likely. The BBC Interview, Face to Face, aired 1959, with transcript published in the above mentioned C.G. Jung Speaking, Princeton University Press, 1977: “I most certainly was characterized by Thinking … and I had a great deal of iNtuition, too. And I had a definite difficulty with Feeling.” So what Jung is saying here is that his Dominant Function is Thinking, his Auxiliary Function is iNtuition and his Repressed Function is Feeling. The fact that INFP is Jung’s least likely type doesn’t bode well for the INTP though. On the other hand, his hunches ( of Nazi Germany) seem more consistent with that of a perceiver than a judger. So now we have INT_?_, and I have to guess a bit at P v. J. His struggle with feeling leads me to believe that his faculties of perception (i.e. intuition) were dominant over his faculties of judgment (i.e thinking).

The Case for ISTP

The manuscript prepared in 1926, published in English as Analytical Psychology: Notes of the Seminar Given in 1925: “As a natural scientist, thinking and sensation were uppermost in me and intuition and feeling were in the unconscious and contaminated by the collective unconscious. [Princeton University Press 1991 edition, p. 69]” I would attribute this as his persona (mask) and to establish the psychoanalytic method as science.

Was Jung an INTJ or INFJ?

Though his feeling preference was likely much less pronounced than his intuition, the most compelling reason for Jung being INFJ rather than an INTJ, would be Jung’s ineluctable attraction to religion and spirituality.  Jung consistently reiterated Feeling as his repressed Function. There is consensus that Wittgenstein and Plato was INFJ, heaven knows why.

Perhaps the reason has its root in psychology. The Myers-Briggs ‘Thinking’ preference is associated with principled and rule-based thought, whereas its counterpart ‘Feeling’ is associated with decision making on the basis of values. At a statistical level, we can analyse patterns in such preferences, and it is not wholly surprising that the Myers-Briggs types associated with the job of programmer are all Thinking preference types (INTJ, INTP, ISTJ, ISTP, ENTP). Conversely, the job of writer is associated with Feeling preference (INFJ, INFP, ENFJ, ENFP – although also INTP and ENTP), as well the job of artist or composer (ISFP, INFJ, ENFP, INFP – although also INTP and ENTP). It should be stressed, those are just statistical tendencies.