Are you sure the starter is trying to engage by this click description and not your compression releases clicking?
Yes, could be starter; could also be more specifically your starter solenoid. I'd start with taking the cover off your solenoid, pulling the plunger, inspecting cleaning it up and reinstalling if it isn't too badly worn. Check for change of condition.
You could pull the starter and have it tested but knowing me, I'd probably just jumper across the terminals to test it to eliminate the motor versus solenoid plunger.
The fact that your getting a reaction (click) when hitting the start button should eliminate that as the source of the problem.