Bayesian estimation of short-time spectral amplitude is one of the most predominant approaches for the enhancement of the noise corrupted speech. The performance of these estimators are usually significantly improved when any perceptually relevant cost function is considered. On the other hand, the recent progress in the phase-based speech signal processing have shown that the phase-only enhancement based on spectral phase estimation methods can also provide joint improvement in the perceived speech quality and intelligibility, even in low SNR conditions. In this paper, to take advantage of both the perceptually motivated cost function involving STSAs of estimated and true clean speech and utilizing the prior spectral phase information, we have derived a phase-aware Bayesian STSA estimator. The parameters of the cost function are chosen based on the characteristics of the human auditory system, namely, the dynamic compressive nonlinearity of the cochlea, the perceived loudness theory and the simultaneous masking properties of the ear. This type of parameter selection scheme results in more noise reduction while limiting the speech distortion. The derived STSA estimator is optimal in the MMSE sense if the prior phase information is available. In practice, however, typically only an estimate of the clean speech phase can be obtained via employing different types of spectral phase estimation techniques which have been developed throughout the last few years. In a blind setup, we have evaluated the proposed Bayesian STSA estimator with different types of standard phase estimation methods available in the literature. Experimental results have shown that the proposed estimator can achieve substantial improvement in performance than the traditional phase-blind approaches.