For quantitative seismic imaging, iterative least-squares reverse time migration is the recommended approach. The existence of an inverse of the forward modeling operator would considerably reduce the number of required iterations. In the context of the extended model, such a pseudoinverse does exist and is built as a weighted version of the adjoint that accounts for the deconvolution, geometric spreading, and uneven illumination. The application of the pseudoinverse Born modeling is based on constant-density acoustic media, which is a limiting factor for practical applications. To consider density perturbations, we have adopted and investigated two approaches. The first one is a generalization of a recent study proposing to recover acoustic perturbations from the angle-dependent response of the pseudoinverse Born modeling operator. This new version is based on the weighted least-squares objective function. The method not only provides more robust results, but it also offers the flexibility to include constraints in the objective function to reduce the parameters’ crosstalk. We also propose an alternative approach based on Taylor expansion that does not require any Radon transform. Numerical examples based on a simple model and the Marmousi2 models using correct and incorrect background models for the variable density pseudoinverse Born modeling verify the effectiveness of the weighted least-squares method when compared with the other two approaches. The Taylor expansion approach appears to contain too many artifacts for successful applicability.