Appearance-based gaze estimation methods have received increasing attention in the field of human-computer interaction (HCI). These methods tried to estimate the accurate gaze point via Convolutional Neural Network (CNN) model, but the estimated accuracy can't reach the requirement of gaze-based HCI when the regression model is used in the output layer of CNN. Given the popularity of button-touch-based interaction, we propose an appearance-based gaze block estimation method, which aims to estimate the gaze block, not the gaze point. In the proposed method, we relax the estimation from point to block, so that the gaze block can be estimated by CNN-based classification instead of the previous regression model. We divide the screen into square blocks to imitate the button-touch interface, and build an eye-image dataset, which contains the eye images labelled by their corresponding gaze blocks on the screen. We train the CNN model according to this dataset to estimate the gaze block by classifying the eye images. The experiments on 6- and 54-block classifications demonstrate that the proposed method has high accuracy in gaze block estimation without any calibration, and it is promising in button-touch-based interaction.