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In this work we present a nonlinear multiscale finite element method combined with local preconditioning for solving compressible Euler equations in conservative variables. The formulations are based on the strategy of separating scales, in which it is the core of the variational multiscale (finite element) methodology. The subgrid scale space is defined using bubble functions that vanish on the boundary of the elements, allowing to use a local Schur complement to define the resolved scale problem. The resulting numerical procedure allows the fine scales to depend on time. The formulation proposed added artificial viscosity isotropically in all scales of the discretization. Due to the fact that, density-based schemes suffer with undesirable effects of low speed flow including low convergence rate and loss of accuracy, local preconditioning is applied to the set of equations in the continuous case. We evaluate the multiscale formulation with local preconditioning in the low Mach number comparing with the non-preconditioned case. The experiments show that density-based schemes combined with local preconditioning yields good results.