Deconvolution of large survey images with millions of galaxies requires to develop a new generation of methods which can take into account a space variant Point Spread Function (PSF) and have to be at the same time accurate and fast. We investigate in this paper how Deep Learning (DL) could be used to perform this task. We employ a U-Net Deep Neural Network (DNN) architecture to learn in a supervised setting parameters adapted for galaxy image processing and study two strategies for deconvolution. The first approach is a post-processing of a mere Tikhonov deconvolution with closed form solution and the second one is an iterative deconvolution framework based on the Alternating Direction Method of Multipliers (ADMM). Our numerical results based on GREAT3 simulations with realistic galaxy images and PSFs show that our two approaches outperforms standard techniques based on convex optimization, whether assessed in galaxy image reconstruction or shape recovery. The approach based on Tikhonov deconvolution leads to the most accurate results except for ellipticity errors at high signal to noise ratio where the ADMM approach performs slightly better, is also more computation-time efficient to process a large number of galaxies, and is therefore recommended in this scenario.