This paper describes a novel framework for compressive sampling (CS) of multichannel signals that are highly dependent across the channels. In this work, we assume few number of sources are generating the multichannel observations based on a linear mixture model. Moreover, sources are assumed to have sparse/compressible representations in some orthonormal basis. The main contribution of this paper lies in 1) rephrasing the CS acquisition of multichannel data as a compressive blind source separation problem, and 2) proposing an optimization problem and a recovery algorithm to estimate both the sources and the mixing matrix (and thus the whole data) from the compressed measurements. A number of experiments on the acquisition of Hyperspectral images show that our proposed algorithm obtains a reconstruction error between 10 dB and 15 dB less than other state-of-the-art CS methods.