For stability analysis of time-varying time-delay systems, it is known that time-derivatives (time shifts in the discrete-time setting) of the Lyapunov-Krasovskii functionals (LKFs) for both internal stability and input-to-state stability (ISS) can be indefinite, which can improve the resulting stability conditions in some cases. However, the non-strict LKFs may be insufficient and inconvenient to use in practice. In this paper, based on the non-strict ISS LKFs for time-varying time-delay systems and with the help of uniformly asymptotically stable (UAS) and uniformly exponentially bounded (UEB) scalar functions, three classes of strict ISS LKFs are constructed such that their time-derivatives are strictly negative definite along the trajectories of the considered system. A general construction of strict ISS LKFs by using the positive definite and uniformly bounded solution to a scalar Lyapunov differential equation is established, which includes the proposed three classes of strict ISS LKFS as special cases. The approaches are also extended to deal with a Lyapunov inequality whose right hand side is indefinite and nonlinear. Both continuous-time and discrete-time systems are considered. The effectiveness of the proposed methods is illustrated by some examples borrowed from the literature. (C) 2019 Elsevier Ltd. All rights reserved.