In an engine starting system, when a first solenoid is energized at a first timing in response to any one of a turning on of a starter switch and an occurrence of an engine restart request, a solenoid actuator shifts a pinion to a ring gear to be engaged with the ring gear. When a second solenoid is energized, a solenoid switch member energizes a motor. A determiner determines a second timing of energization of the second solenoid after the first timing so that a first delay time from the first timing to the second timing when the first timing is responsive to the turning on of the starter switch is different from a second delay time from the first timing to the second timing when the first timing is responsive to the occurrence of the engine restart request.