Lattice distortions, i.e. small atomic displacements away from the average lattice, are linked to a number of functional and mechanical properties of concentrated metallic alloys, particularly their yield strength. Here we develop an elastic model of lattice distortions where every atom is modeled as an Eshelby inclusion in a homogeneous elastic matrix. Local environment effects are included by considering fluctuating anisotropic eigenstrain tensors associated to the inclusions. The model is tested on several concentrated alloys, face-centered cubic (FCC) AlMg and FeNiCr alloys, as well as a fictitious body-centered cubic (BCC) binary alloy to study systematically and independently the effects of a size and an elastic modulus mismatch between the constituents. The elastic model predicts lattice distortions well when the size and elastic modulus mismatches are typically less than 5% and 25%, respectively. Interestingly, we find that when the size mismatch is small, as in FeNiCr alloys, the lattice distortion is dominated by the fluctuations of the dilatational eigenstrains rather than their average value as often assumed in elastic models of concentrated alloys. Moreover, models usually assume homogeneous elastic constants, while the limit obtained here of 25% is often exceeded in concentrated alloys, particularly with a BCC structure.