Recently, some methods have been proposed for filtering multi-coil MRI acquisitions with correlation between coils. Those methods are based on statistical models of noise to develop a Linear Minimum Mean Square Error (LMMSE) filter. The advantage of LMMSE-based filters stems from their simplicity and robustness. However, they exhibit some drawbacks: their performance strongly depends on the underlying statistical model and on the way the local moments are estimated. The first problem can be avoided when considering effective values provided by recent studies on the models of noise in multi-coil systems with correlation between coils. However, the local moments are estimated in square neighborhoods which can include different kinds of tissues. Thus, the local variance is biased towards upper values, which results in an inaccurate estimate in regions close to tissue boundaries. In this work we propose to overcome this problem by introducing an anisotropic diffusion step in the LMMSE estimate for correlated multi-coil systems which improves the estimation of the signal in regions where other LMMSE methods fail. Results demonstrate the better behavior in different noisy scenarios.