Rapid and continuous advancements in remote sensing technology have resulted in finer resolutions and higher acquisition rates of hyperspectral images (HSIs). These developments have triggered a need for new processing techniques brought about by the confined power and constrained hardware resources aboard satellites. This article proposes two novel lossless and near-lossless compression methods, employing our recent seed generation and quadrature-based square rooting algorithms, respectively. The main advantage of the former method lies in its acceptable complexity utilizing simple arithmetic operations, making it suitable for real-time onboard compression. In addition, this near-lossless compressor could be incorporated for hard-to-compress images offering a stabilized reduction at nearly 40% with a maximum relative error of 0.33 and a maximum absolute error of 30. Our results also show that a lossless compression performance, in terms of compression ratio, of up to 2.6 is achieved when testing with hyperspectral images from the Corpus dataset. Further, an improvement in the compression rate over the state-of-the-art k2-raster technique is realized for most of these HSIs by all four variations of our proposed lossless compression method. In particular, a data reduction enhancement of up to 29.89% is realized when comparing their respective geometric mean values.