What is the meaning of "gut " in this context?

My first instinct is to call Coop and tell him about it. I have no doubt he’d drop everything and drive into Manhattan for the second day in a row just to assure me that nothing about this is my fault. But I’m not sure I want to see Coop on consecutive days. It would be the first time that happened since Pine Cottage and the morning after, and it’s not an experience I long to repeat.
I text him instead, trying to keep it casual.
Call me when you get a chance. No rush. Nothing important.
But my gut tells me it is important. Or at least it has the potential to be. If it wasn’t important, why did I wake up thinking about it? Why is my next thought to call Jeff just to hear his voice, even though I know he’s in court, his cell phone switched off and shoved into the depths of his briefcase?