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This paper introduces a novel technique for signal-to-noise ratio (SNR) estimation for scenarios where two closely-spaced microphones are available. The proposed technique utilizes the real and imaginary parts of the coherence function between the input signals to estimate the SNR without assuming prior knowledge of the noise statistics. The corresponding dual-microphone speech enhancement algorithm utilizes a Wiener filter as a gain function constructed using the SNR values computed by the coherence function. Since the proposed SNR estimation technique does not require access to noise statistics, it can be applied in situations where interfering speakers are present. An adaptive speech reception threshold (SRT) test was used to assess the intelligibility of speech processed by the proposed algorithm in scenarios where one or two interfering talkers were present in anechoic and reverberant conditions. Intelligibility listening tests were conducted with both normal-hearing (NH) and cochlear implant (CI) listeners. Results revealed significant improvements in intelligibility and quality over a (baseline) fixed directional algorithm and a well-established beamformer algorithm. In a nearly anechoic room with competing talkers, the improvement in SRT obtained relative to the directional microphone ranged from 5-10 dB, while the improvement obtained by the beamformer was about 2 dB. In reverberant environments, the improvement in SRT remained high (4-7 dB) at T60 = 220 ms, and decreased to 1-2 dB at T60 = 465 ms. Overall, the proposed algorithm provided significant benefits in intelligibility in anechoic and mildly reverberant environments making it suitable for hearing aid and cochlear implant applications.