The study of input-output stability of reset control systems with time-varying delay is addressed in this work. The time-varying function that defines the delay is assumed to be bounded on magnitude and variation. This approach also covers the particular case of constant time delay, but it is studied separately to obtain less conservative results. After proposing a convenient loop transformation, the stability analysis is performed by means of the integral quadratic constraint framework. Then by applying the Kalman-Yakubovich-Popov lemma, easily checkable conditions in form of linear matrix inequalities are obtained. To conclude, some numerical examples are provided illustrating the proposed criteria.