Two adaptive algorithms are presented for robust time delay estimation (TDE) in acoustic environments with a large amount of background noise and reverberation. Recently, an adaptive eigenvalue decomposition (EVD) algorithm has been developed for TDE in highly reverberant acoustic environments. In this paper, we extend the adaptive EVD algorithm to noisy and reverberant acoustic environments, by deriving an adaptive stochastic gradient algorithm for the generalized eigenvalue decomposition (GEVD) or by prewhitening the noisy microphone signals. We have performed simulations using a localized and a diffuse noise source for several SNRs, showing that the time delays can be estimated more accurately using the adaptive GEVD algorithm than using the adaptive EVD algorithm. In addition, we have analyzed the sensitivity of the adaptive GEVD algorithm with respect to the accuracy of the noise correlation matrix estimate, showing that its performance may be quite sensitive, especially for low SNR scenarios.