Due to their noise-like features, SAR images are difficult to acquire with compressed sensing techniques. However, some parts of the images, typically associated to man-made structures, are compressible and we investigate two techniques exploiting that information to allow a compressive acquisition of the whole image. These techniques result in a significant enhancement of the image quality compared to classical compressed sensing. Moreover, compared to classical sampling and quantisation of the SAR raw data, they allow a significant reduction of bitrate with a limited increase of the distortion. However, their efficiency depends strongly on the presence of compressible parts in the image.