Speech Intelligibility Prediction (SIP) algorithms are becoming popular tools within the development and operation of speech processing devices and algorithms. However, many SIP algorithms require knowledge of the underlying clean speech; a signal that is often not available in real-world applications. This has led to increased interest in non-intrusive SIP algorithms, which do not require clean speech to make predictions. In this paper we investigate the use of Convolutional Neural Networks (CNNs) for non-intrusive SIP. To do so, we utilize a CNN architecture that shows similarities to existing SIP algorithms, in terms of computational structure, and which allows for easy and meaningful visualization and interpretation of trained weights. We evaluate this architecture using a large dataset obtained by combining datasets from the literature. The proposed method shows high prediction performance when compared with four existing intrusive and non-intrusive SIP algorithms. This demonstrates the potential of deep learning for speech intelligibility prediction.