Partial state estimation has numerous applications in practice. Nevertheless, designing delay-dependent functional observers (FOs) for systems with unknown time delays is rigorous and still an open dilemma. This paper addresses the problem for linear time-invariant systems with state time-varying delays. The delay is assumed to be bounded in an interval with a bounded derivative. A sliding mode FO structure that is robust against the delay uncertainties is established to this aim. The structure employs an auxiliary delay function that can be defined based on the existing knowledge on the actual delay values. Delay-dependent sufficient conditions for the stability of the observer are obtained using the Lyapunov Krasovskii approach, and are expressed in terms of a linear matrix inequality and two rank conditions. The delay-free observer structure is additionally studied and the necessary and sufficient conditions for its stability are obtained. Two descriptive numerical examples and simulation results demonstrate the design procedure and emphasize the effectiveness of the proposed observer design algorithm.