gut feeling

a gut feeling AND a gut instinct (an instinct or intuition; an immediate or basic feeling or reaction without a logical rationale) — чутье, интуиция

Example 1: Her gut feeling was that he was lying and, sure enough, he was. — Чутье подсказывало ей, что он лжет. Собственно, так оно и было.
Example 2: But being a good doctor requires sticking with the best medical evidence, even if it contradicts your personal experience. We need to distinguish between gut feeling and testable knowledge, between hunches and empirically tested evidence. (Robert Burton)

have a gut feeling (about sth / that) (sense something about a person or a situation, without knowing why) — нутром чуять; предчувствовать

I have a gut feeling (that) ~ чует мое сердце; нутром чую, что…

Example: He had a gut feeling that his wife was cheating on him.

gut-level — интуитивный, на уровне интуиции
...using "gut-level leadership" principles

trust one's gut — доверять своей интуиции, довериться интуиции

Example: Trust your gut! If you aren’t sure as to whether or not a boundary is being broken, stop thinking and start feeling. At times, our guts have better listening skills than our ears. If you can feel a visceral reaction to the conversation at hand, you can be pretty sure that something isn’t right.

Origin:
Probably derived from the fact that many people experience emotions and intuitive feelings as being centred on, or having a strong effect on, the stomach area, which is also called the gut. Interestingly, the nervous system's second biggest network of closely-interconnected neurones, after the brain, is located in this area of the body.(EnglishClub)

Related vocabulary:
[feel in one's bones]
[by the seat of one's pants]