We propose a method -- a quantum time mirror (QTM) -- for simulating a partial time-reversal of the free-space motion of a nonrelativistic quantum wave packet. The method is based on a short-time spatially-homogeneous perturbation to the wave packet dynamics, achieved by adding a nonlinear time-dependent term to the underlying Schrödinger equation. Numerical calculations, supporting our analytical considerations, demonstrate the effectiveness of the proposed QTM for generating a time-reversed echo image of initially localized matter-wave packets in one and two spatial dimensions. We also discuss possible experimental realizations of the proposed QTM.