This note presents a new method to design adaptive feedback controller for nonlinear time-delay systems. First, by using the LaSalle-Razumikhin theorem, a sufficient condition is derived that ensures the convergence of a part of the solution with stability for a class of functional differential equations. Then, using this condition, the adaptive stabilization problem is solved for nonlinear time-delay systems. Moreover, it is shown that for a class of nonlinear time-delay systems with triangular structure the adaptive stabilizing controller can be obtained by recursively constructing the Lyapunov-Razumikhin function. It will be shown that the provided recursive design approach, which is obviously motivated by the typical backstepping method, is not a trivial extension of the existing design method.