This paper proposes a novel method for sharpening the 20 m bands of the multispectral images acquired by the Sentinel-2 (S2) constellation. We formulate the S2 sharpening as an inverse problem and solve it using an unsupervised convolutional neural network (CNN), called S2UCNN. The proposed method extends the deep image prior provided by a CNN structure with S2 domain knowledge. We incorporate a modulation transfer function-based degradation model as a network layer. We add the 10 m bands to both the network input and output to take advantage of the multitask learning. Experimental results with a real S2 dataset show that the proposed method outperforms the competitive methods on reduced-resolution data and gives very high quality sharpened image on full-resolution data.