The starter relay does that. You don't hear it when the starter works..and makes noise.
An aside: For working on ignition wiring..maybe chasing down start/run ballast resistor problems, remove the 1/4" screw that holds the starter solenoid wire onto the relay. Be careful...it's close to that big stud that's hot. Touch the two and the starter fires. Again, the reason for the safeguards: Car in neutral, E brake on, coil grounded....all that kind'a stuff. With the starter 'go' wire disconnected, you can meter any start/run IG1/IG2 wiring issues you might have and leave the starter out of it.
Aside #2: Don't leave the key 'on' for a long period of time. That's not a normal situation for the car to, say, have the coil sinking power constantly. Things get hot and can get damaged.
