Estimation of the time delay of arrival (TDOA) problem is important to acoustic source localization. The TDOA estimation problem is defined as finding the relative delay between several microphone signals for the direct sound. To estimate TDOA, the generalized cross-correlation (GCC) method is the most frequently used, but it has a disadvantage in terms of reverberant environments. In order to overcome this problem, the adaptive eigenvalue decomposition (AED) method has been developed, which estimates the room transfer function and finds the direct-path delay. However, the algorithm does not take into account the fact that the room transfer function is a sparse channel, and so sometimes the estimated transfer function is too dense, resulting in failure to exact direct-path and delay. In this paper, an enhanced AED algorithm that makes use of a proportionate step-size control and a direct-path constraint is proposed instead of a constant step size and the L2-norm constraint. The simulation results show that the proposed algorithm has enhanced performance as compared to both the conventional AED method and the phase-transform (PHAT) algorithm.