This paper proposes an improved estimation of the reachable set (RS) analysis in linear systems with polytopic uncertainties, peak-bounded inputs and time-varying delay. Inspired by past literature, Lyapunov-Krasovskii's functionals are dealt for treating the time-delay and bounding analysis effectively. So, the proposed method focuses on Lyapunov-Krasovskii's functionals via various time-delay conditions for linear systems. Based on the Lyapunov method, some integral inequalities, useful zero equalities, and the augmented zero equality approach are introduced. The results are expressed in terms of linear matrix inequalities, which are easy to get optimized solutions for obtaining guaranteed minimum RS of system dynamics. Finally, two numerical examples are shown to judge that the proposed estimation method can lead to less conservative results.