In this paper, a new stability criterion for linear time-delay systems of neutral type is presented. By virtue of the well-known Krasovskii theorem, the system is exponentially stable, if and only if there exists a functional which (i) has a negative definite derivative along the solutions of the system, and (ii) admits a quadratic lower bound, of course under an assumption that the difference operator corresponding to the system is stable. We modify the second condition in the following manner: The functional is required to admit a quadratic lower bound only on the special set of initial functions in our stability result instead of the set of all appropriate functions in the Krasovskii theorem. This special set consists of the functions satisfying a Razumikhin-type inequality and a similar inequality on the derivative. Basing on such modification and staying within the framework of the functionals with a given derivative, we suggest a methodology for the stability analysis, which is described and tested on the examples in the case of a scalar equation. Unlike most of recent results, we use the functional with a derivative prescribed just as a negative definite quadratic form of the "current" state of a system, what does not impose restrictions on the approach. (C) 2019 Elsevier Ltd. All rights reserved.