To have a gut feeling means to have an intuition about something or somebody. It refers to the fact that one feels something uneasy in the region of the guts when you eat something that is not quite agreeable to your digestive system.
1. Mrs. Bali had a gut feeling that her daughter was not exactly telling the truth about her exam results.
2. The pilot had a gut feeling that they would fly into rough weather if they took off now.
3. The coach had a gut feeling that the rival team's coach would substitute Jacob for Paul in the second half of the game.