This paper presents a convolutional neural network-(CNN-) based pupil center detection method for a wearable gaze estimation system using infrared eye images. Potentially, the pupil center position of a user's eye can be used in various applications, such as human-computer interaction, medical diagnosis, and psychological studies. However, users tend to blink frequently; thus, estimating gaze direction is difficult. The proposed method uses two CNN models. The first CNN model is used to classify the eye state and the second is used to estimate the pupil center position. The classification model filters images with closed eyes and terminates the gaze estimation process when the input image shows a closed eye. In addition, this paper presents a process to create an eye image dataset using a wearable camera. This dataset, which was used to evaluate the proposed method, has approximately 20,000 images and a wide variation of eye states. We evaluated the proposed method from various perspectives. The result shows that the proposed method obtained good accuracy and has the potential for application in wearable device-based gaze estimation.