In this paper, we introduce a variational Bayesian algorithm (VBA) for image blind deconvolution. Our VBA generic framework incorporates smoothness priors on the unknown blur/image and possible affine constraints (e.g., sum to one) on the blur kernel, integrating the VBA within a neural network paradigm following an unrolling methodology. The proposed architecture is trained in a supervised fashion, which allows us to optimally set two key hyperparameters of the VBA model and leads to further improvements in terms of resulting visual quality. Various experiments involving grayscale/color images and diverse kernel shapes, are performed. The numerical examples illustrate the high performance of our approach when compared to state-of-the-art techniques based on optimization, Bayesian estimation, or deep learning.