In this paper, controllability of the linear discrete-time systems $(A,B,\Omega­):x_{k+1}=Ax_k+Bu_k,\ x_k\in X, u_k\in\Omega$­, is studied, where $X$ is a Banach space and the control set ­ is assumed to be a cone in a Banach space $U$. Some criteria for approximate controllability are given. The case where the operator $A$ is compact is examined in detail by using the spectral decomposition of the state space $X$. As a result, a criterion for approximate controllability of $(A,B,\Omega­)$ is obtained without imposing a restrictive condition that the system with no control constraints $(A,B,U)$ is exactly controllable. The obtained results are then applied to consider the problem of controllability for linear functional differential equations with positive controls. Some necessary and sufficient conditions of approximate controllability to the state space $\mathbf{R}^n\times L_p$ are presented and some illustrating examples are given.