Cochlear implants (CI) are implantable electronic prostheses that aim to restore communication in people with severe to profound hearing loss. This is achieved by transforming the acoustic signals into electrical stimuli and applying them directly to the cochlea through a set of electrodes. Despite its effectiveness under optimal environmental conditions, its performance is severely degraded in the presence of noise. This work proposes an envelope-based minimum-variance distortionless response (MVDR) approach for multichannel noise reduction in CI applications. The original constraint in the conventional time-domain MVDR beamformer is modified to ensure that the second-order statistical moments of the processed and original speech-only envelopes are equivalent. Assuming strict computational limitations, as those found in commercial multielectrode systems, a low computational-cost particular case of the proposed method results in a power-based MVDR that has a semianalytical solution. The resulting algorithm provides a fast calculation method for its optimal coefficients without losing significant performance compared to its general form. Results obtained from computational simulations and objective criteria indicate higher intelligibility levels achieved by the proposed methods compared to the conventional time-domain MVDR beamformer. This evidence is corroborated by psychoacoustic experiments with ten normal-hearing volunteers listening to vocoded speech, and one CI user employing a research interface.