A sum-of-squares (SOS) approach is proposed to estimate the domain of attraction (DoA) of regulated nonlinear polynomial time-delay systems subject to input constraints. It will be shown that the joint use of semialgebraic geometric arguments and semidefinite programming methods provides an efficient tool for Lyapunov-based control system analysis and design for this class of nonlinear delayed systems. Specifically, a two-step procedure is presented which consists of deriving first a Lyapunov-Krasovskii functional/controller pair complying with the prescribed constraints and computing then an estimate of the DoA.