To reduce time and costs in the development process of noise reduction algorithms, an objective intelligibility measure is crucial. Such a measure has to show high correlation with speech intelligibility determined by real listening experiments. In the past several measures were found that perform reliable in a particular scenario when only the spectral amplitude of a noisy signal is modified. Recent studies demonstrate the positive impact of a phase modification in a single-channel speech enhancement showing improved speech intelligibility while conventional methods relying on amplitude-only modification are known for reduced intelligibility. Further, another recent study shows that a distortion metric defined on the spectral phase outperforms state-of-the-art quality metrics when used in phase-aware speech enhancement. This raises two questions we account for in this work; First, to study the reliability of the existing intelligibility measures in predicting the performance of the phase-aware methods, and second to investigate candidates for new phase-aware instrumental metrics and evaluate their reliability in terms of intelligibility prediction. Our objective and subjective evaluations demonstrate that CSII-based and STOI as well as the proposed phase-aware metrics perform as reliable speech intelligibility estimators following the subjective results.