We consider a local version of the pole assignment problem for linear discrete time-varying systems. Our aim is to obtain sufficient conditions to place the Lyapunov spectrum of the closed-loop system in an arbitrary position within some neighborhood of the Lyapunov spectrum of the free system using an appropriate time-varying linear feedback. Moreover, we assume that the norm of the matrix of linear feedback should be bounded from above by the distance between these two spectra with some constant multiplier. We prove that diagonalizability, Lyapunov regularity, and stability of the Lyapunov spectrum each separately are the required sufficient conditions provided that the open-loop system is uniformly completely controllable.