This paper is concerned with the reachable set estimation problem for discrete-time linear systems with multiple constant delays and bounded peak inputs. The objective is to check whether there exists a bounded set that contains all the system states under zero initial conditions. First, delay-dependent conditions for the solvability of the addressed problem are derived by employing a novel Lyapunov–Krasovskii functional. The obtained conditions are expressed in terms of matrix inequalities, which are linear when only one scalar variable is fixed. On the basis of these conditions, an ellipsoid containing the reachable set of the considered system is obtained. An approach for determining the smallest ellipsoid is also provided. Second, the approach and results developed in the first stage are generalized to the case of systems with polytopic parameter uncertainties, and delay-dependent conditions are given in the form of relaxed matrix inequalities. Finally, two numerical examples are provided to demonstrate the effectiveness of the proposed methods. Copyright © 2013 John Wiley & Sons, Ltd.