We consider a novel approach to the information bottleneck problem where the goal is to perform compression of a noisy signal, while retaining a significant amount of information about a correlated auxiliary signal. To facilitate analysis, we cast compression with side information as an optimization problem involving an information measure, which for jointly Gaussian random variables equals the classical mutual information. We provide closed form expressions for locally optimal linear compression schemes; in particular, we show that the optimal solutions are of the form of the product of an arbitrary full-rank matrix and the left eigenvectors corresponding to smallest eigenvalues of a matrix related to the signals' covariance matrices. In addition, we study the influence of the sparsity level of the Bernoulli-Gaussian noise on the compression rate. We also highlight the similarities and differences between the noisy bottleneck problem and canonical correlation analysis (CCA), as well as the Gaussian information bottleneck problem.