This paper describes a novel framework for compressive sampling of multichannel signals that are highly correlated across the channels. In this work, we assume few number of independent 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 rephrasing the compressed sampling of multichannel data as the compressive source separation problem by knowing the mixture parameters. A number of simulations measure the performance of our recovery algorithm. Comparing to the classical CS scheme -which recovers data of all channels separately- ours indicates a significant reduction in both, the number of measurements to be sent and the complexity of the decoding algorithm (i.e., scaling by the number of sources rather than the number of channels). We demonstrate an application of our scheme in acquisition of the Hyperspectral images. Our algorithm proposes an accurate, low cost recovery from small number of the transmitted measurements that are taken by sensors with very low spatial resolution.