In this contribution, the estimates for the response of time-delay systems with nonlinear homogeneous right-hand side of degree strictly greater than one are constructed. The existing results obtained via the Lyapunov--Razumikhin approach are reminded. Their proofs, revisited in the appendix, lead to explicit expressions of the involved constants. Based on a recently introduced Lyapunov–Krasovskii functional and known estimates of the domain of attraction, we present new estimates of the system response. Using the same Lyapunov function in the Lyapunov–Razumikhin approach and in construction of the Lyapunov–Krasovskii functional, we compare both approaches and discuss the illustrative examples.