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Deep learning based image denoising methods have been recently popular due to their improved performance. Traditionally, these methods are trained in a supervised manner, requiring a set of noisy input and clean target image pairs. More recently, self-supervised approaches have been proposed to learn denoising from only noisy images. These methods assume that noise across pixels is statistically independent, and the underlying image pixels show spatial correlations across neighborhoods. These methods rely on a masking approach that divides the image pixels into two disjoint sets, where one is used as input to the network while the other is used to define the loss. However, these previous self-supervised approaches rely on a purely data-driven regularization neural network without explicitly taking the masking model into account. In this work, building on these self-supervised approaches, we introduce Noise2Inpaint (N2I), a training approach that recasts the denoising problem into a regularized image inpainting framework. This allows us to use an objective function, which can incorporate different statistical properties of the noise as needed. We use algorithm unrolling to unroll an iterative optimization for solving this objective function and train the unrolled network end-to-end. The training paradigm follows the masking approach from previous works, splitting the pixels into two disjoint sets. Importantly, one of these is now used to impose data fidelity in the unrolled network, while the other still defines the loss. We demonstrate that N2I performs successful denoising on real-world datasets, while better preserving details compared to its purely data-driven counterpart Noise2Self.