Drivers do not need to supervise the L3 automated driving system but have to resume dynamic driving tasks when necessary, where the takeover request system plays a crucial role in alerting drivers to take over control. Previous studies have investigated the feasibility of the two-stage takeover system, which divides the takeover request system into monitoring requests and takeover requests. However, whether it will affect drivers’ situation awareness and takeover performance if the monitoring request is issued multiple times without the takeover request has not been fully considered. This research conducted a driving simulation study to investigate the effect of multiple monitoring requests on participants’ vigilance to monitoring requests and readiness for takeover requests by measuring their eye movement and takeover performance. Forty-two participants experienced three kinds of takeover conditions: no monitoring request, once monitoring request, and four monitoring requests. Simultaneously, the eye movement and takeover performance of participants were recorded and analyzed. Results showed that participants gained sufficient situation awareness in the monitoring request phase and showed better takeover performance in the two-stage takeover system than the single-stage takeover system. However, participants’ vigilance to the monitoring request and readiness for the takeover request decreased after experiencing multiple monitoring requests. Specifically, 62% and 59% of participants put their hands on the steering wheel during the monitoring request phase when experiencing the first monitoring request, which decreased to 35% after experiencing four monitoring requests. Moreover, after experiencing multiple monitoring requests, more participants shifted their attention to non-driving-related tasks after first observing the road and with a reduced percentage of time in the road center. These findings suggest that the monitoring request should be more reasonable to avoid reducing the vigilance and readiness of drivers to take advantage of the two-stage takeover system effectively.