This paper studies the qualitative behavior of solutions of linear discrete systems with constant coefficients and a constant time delay under the influence of a control function. In particular, it deals with a nontrivial task of transferring an arbitrary solution to several prescribed positions (trajectory controllability), being more general and different from the problem of relative controllability frequently solved in the literature. A Kalman-type controllability criterion is proved and, in addition, previous results are discussed and illustrative examples are given.