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Recent studies show that deep learning (DL) based MRI reconstruction outperforms conventional methods, such as parallel imaging and compressed sensing (CS), in multiple applications. Unlike CS that is typically implemented with pre-determined linear representations for regularization, DL inherently uses a non-linear representation learned from a large database. Another line of work uses transform learning (TL) to bridge the gap between these two approaches by learning linear representations from data. In this work, we combine ideas from CS, TL and DL reconstructions to learn deep linear convolutional transforms as part of an algorithm unrolling approach. Using end-to-end training, our results show that the proposed technique can reconstruct MR images to a level comparable to DL methods, while supporting uniform undersampling patterns unlike conventional CS methods. Our proposed method relies on convex sparse image reconstruction with linear representation at inference time, which may be beneficial for characterizing robustness, stability and generalizability.